Explanation:
Okay, so I need to find z_{2002} where each term is defined by iterating the function F(z) = (z + i)/(z - i), starting from z_0 = 1/137 + i. Hmm, let me think about how to approach this.

First, since the function is being iterated, maybe there's a pattern or periodicity in the sequence z_n? If I can find the period, then I can figure out what z_{2002} is by finding the remainder when 2002 is divided by the period. But how do I find the period?

Alternatively, maybe I can find a closed-form expression for z_n. Let me try to analyze the function F(z). Let me write it down again:

F(z) = (z + i)/(z - i)

This looks like a Möbius transformation. Möbius transformations can sometimes be analyzed by finding their fixed points or by conjugating them to a simpler form, like a rotation or translation. Maybe I can find the fixed points of F(z)?

To find fixed points, set F(z) = z:

(z + i)/(z - i) = z

Multiply both sides by (z - i):

z + i = z(z - i)

Expand the right side:

z + i = z^2 - i z

Bring all terms to one side:

z^2 - i z - z - i = 0

Combine like terms:

z^2 - (1 + i) z - i = 0

Now, solve this quadratic equation for z. Let me compute the discriminant:

D = [-(1 + i)]^2 - 4 * 1 * (-i) = (1 + 2i + i^2) + 4i = (1 + 2i - 1) + 4i = 2i + 4i = 6i

Wait, discriminant is 6i? That's a complex number. Let me check my calculation again.

Original equation after moving terms:

z^2 - (1 + i) z - i = 0

Discriminant D = [-(1 + i)]^2 - 4 * 1 * (-i) = (1 + i)^2 + 4i

Compute (1 + i)^2: 1 + 2i + i^2 = 1 + 2i -1 = 2i. Then add 4i: 2i + 4i = 6i. Yes, that's correct. So D = 6i.

So solutions are:

z = [ (1 + i) ± sqrt(6i) ] / 2

Hmm, sqrt(6i)... Let me compute sqrt(i). Since i = e^{iπ/2}, so sqrt(i) = e^{iπ/4} = (1 + i)/√2. Therefore sqrt(6i) = sqrt(6) * e^{iπ/4} = sqrt(6)*(1 + i)/√2 = sqrt(3)*sqrt(2)*(1 + i)/√2 = sqrt(3)(1 + i). Wait, let me check:

sqrt(6i) = sqrt(6) * sqrt(i) = sqrt(6) * (1 + i)/√2 = sqrt(6)/sqrt(2) * (1 + i) = sqrt(3) * (1 + i). Yes, that's correct. So sqrt(6i) = sqrt(3)(1 + i). Therefore:

z = [ (1 + i) ± sqrt(3)(1 + i) ] / 2 = [ (1 + i)(1 ± sqrt(3)) ] / 2

Therefore, the fixed points are:

z = [ (1 + i)(1 + sqrt(3)) ] / 2 and z = [ (1 + i)(1 - sqrt(3)) ] / 2

Hmm, interesting. But I don't know if fixed points are directly helpful here. Maybe I need to consider the transformation properties. Alternatively, perhaps I can find an expression for z_n in terms of z_0 by iterating F.

Alternatively, maybe I can find an invariant or a transformation that simplifies the iteration. Let me think. Möbius transformations can be conjugated to translations or rotations if they have certain properties. Let me check if F(z) is a rotation or something else.

Alternatively, maybe I can use complex analysis. Let me consider writing z in terms of real and imaginary parts, but that might get messy. Alternatively, maybe consider using substitution. Let me try to find a substitution that linearizes the transformation.

Let me consider the function F(z) = (z + i)/(z - i). Let me try to find a substitution w(z) such that F(z) becomes a simpler function in terms of w. For example, if I can find a function w(z) such that w(F(z)) = w(z) + c or something like that, which would make the iteration easier.

Alternatively, since Möbius transformations can be represented by matrices, maybe I can find the matrix corresponding to F(z) and then find its powers. Let me recall that a Möbius transformation f(z) = (az + b)/(cz + d) corresponds to the matrix [[a, b], [c, d]], up to scalar multiples. So for F(z) = (z + i)/(z - i), the matrix would be [[1, i], [1, -i]]. Let me check:

Yes, because if you write f(z) = (az + b)/(cz + d), then a=1, b=i, c=1, d=-i. So the matrix is [[1, i], [1, -i]]. Now, if I can diagonalize this matrix or find its eigenvalues, then I can compute its powers more easily.

So let me find the eigenvalues of the matrix M = [[1, i], [1, -i]]. The eigenvalues λ satisfy det(M - λI) = 0:

|1 - λ    i        |
|1     -i - λ|  = 0

Compute determinant: (1 - λ)(-i - λ) - (i)(1) = 0

Expand:

(1 - λ)(-i - λ) - i = 0

Multiply out (1 - λ)(-i - λ):

= -i(1 - λ) - λ(1 - λ) = -i + iλ - λ + λ²

Then subtract i:

Total expression: -i + iλ - λ + λ² - i = λ² + iλ - λ - 2i = 0

So equation is λ² + (i - 1)λ - 2i = 0

Let me solve this quadratic equation for λ:

λ = [ -(i - 1) ± sqrt( (i - 1)^2 + 8i ) ] / 2

First compute (i - 1)^2:

(i - 1)^2 = (-1 + i)^2 = 1 - 2i + i² = 1 - 2i -1 = -2i

Then add 8i: -2i + 8i = 6i

So sqrt(6i) again. Wait, same as before. So sqrt(6i) is sqrt(3)(1 + i). Therefore:

λ = [ -(i - 1) ± sqrt(3)(1 + i) ] / 2

Simplify -(i - 1) = 1 - i. Therefore:

λ = [1 - i ± sqrt(3)(1 + i)] / 2

So the eigenvalues are:

λ₁ = [1 - i + sqrt(3)(1 + i)] / 2

and

λ₂ = [1 - i - sqrt(3)(1 + i)] / 2

Hmm, this seems complicated. Maybe there's a better way. Alternatively, perhaps I can find a function that conjugates F(z) to a simpler transformation. For example, if I can find a function w(z) such that w(F(z)) = w(z) + c, then iterating F would correspond to adding c each time. Let me try to find such a function.

Alternatively, since Möbius transformations can be conjugated to translations if they have a fixed point. Wait, if the transformation has two fixed points, then it can be conjugated to a translation. Let me recall that. If a Möbius transformation has two fixed points, then it's conjugate to a translation. So maybe I can conjugate F(z) to a translation, which would make the iteration easier.

So let me try that. Let me first find the fixed points again. Earlier, I found that the fixed points are z = [ (1 + i)(1 ± sqrt(3)) ] / 2. Let me denote them as α and β. Let me compute them numerically to get a sense.

First, compute sqrt(3) ≈ 1.732. So:

For α: [ (1 + i)(1 + sqrt(3)) ] / 2 ≈ [ (1 + i)(2.732) ] / 2 ≈ (2.732 + 2.732i)/2 ≈ 1.366 + 1.366i

For β: [ (1 + i)(1 - sqrt(3)) ] / 2 ≈ [ (1 + i)(-0.732) ] / 2 ≈ (-0.732 - 0.732i)/2 ≈ -0.366 - 0.366i

So the fixed points are approximately 1.366 + 1.366i and -0.366 - 0.366i. Let me denote them as α and β.

Now, if I can find a Möbius transformation that maps these fixed points to 0 and ∞, then the transformation would conjugate F(z) to a translation. Let me recall that if we have two fixed points α and β, then the transformation that maps α to 0 and β to ∞ is given by w(z) = (z - α)/(z - β). Then, the conjugated transformation would be w(F(z)) = w(z) + c for some constant c. Let me check.

Let me define w(z) = (z - α)/(z - β). Then, let me compute w(F(z)):

First, compute F(z) = (z + i)/(z - i). Then, w(F(z)) = [F(z) - α]/[F(z) - β] = [ (z + i)/(z - i) - α ] / [ (z + i)/(z - i) - β ]

Let me compute numerator and denominator:

Numerator: [ (z + i) - α(z - i) ] / (z - i) = [ z + i - α z + α i ] / (z - i) = [ (1 - α) z + i(1 + α) ] / (z - i)

Denominator: [ (z + i) - β(z - i) ] / (z - i) = [ z + i - β z + β i ] / (z - i) = [ (1 - β) z + i(1 + β) ] / (z - i)

Therefore, w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

But since w(z) = (z - α)/(z - β) = [ z - α ] / [ z - β ] = [ z - α ] / [ z - β ]

So if I want w(F(z)) = w(z) + c, then:

[ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] = [ z - α ] / [ z - β ] + c

This seems complicated. Maybe there's a better way. Alternatively, since we know that the transformation F(z) has fixed points α and β, then the conjugated transformation should be a translation. Let me recall that if we have a Möbius transformation with fixed points α and β, then it can be conjugated to a translation by the map w(z) = (z - α)/(z - β). Let me check this.

Let me denote the conjugated transformation as T(w) = w + c. Then, the composition would be:

w(F(z)) = T(w(z)) = w(z) + c

So, if I can find c such that this holds, then the iteration becomes w_{n+1} = w_n + c, which is easy to solve.

Let me try to compute this. Let me denote w(z) = (z - α)/(z - β). Then, we need:

w(F(z)) = w(z) + c

Let me compute w(F(z)):

As above, we have:

w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

But also, w(z) + c = [ (z - α)/(z - β) ] + c = [ (z - α) + c(z - β) ] / (z - β) = [ (1 + c) z - α - c β ] / (z - β)

Therefore, equating the two expressions:

[ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] = [ (1 + c) z - α - c β ] / (z - β)

Cross-multiplying:

[ (1 - α) z + i(1 + α) ] * (z - β) = [ (1 - β) z + i(1 + β) ] * [ (1 + c) z - α - c β ]

This equation must hold for all z (except the fixed points). Therefore, the coefficients of corresponding powers of z must be equal. This seems complicated, but maybe we can find c by choosing specific values or by using the fact that α and β are fixed points.

Alternatively, since α and β are fixed points of F(z), then F(α) = α and F(β) = β. Therefore, w(α) = 0 and w(β) = ∞. Then, if we consider the transformation T(w) = w + c, then applying T to w(F(z)) should give us w(z) + c. But since F(z) maps α to α, then w(F(α)) = w(α) = 0, and T(w(α)) = 0 + c = c. But also, w(F(α)) = w(α) + c = 0 + c = c. So that's consistent. Similarly for β, but since w(β) is ∞, adding c would still be ∞. So maybe this approach is not giving me the value of c directly.

Alternatively, maybe I can use the fact that the transformation F(z) is conjugate to a translation. Let me recall that if we have a Möbius transformation with two fixed points, then it's conjugate to a translation. The translation amount can be determined by the derivative at the fixed point or something else. Alternatively, since we have the matrix representation, maybe we can find the trace or something.

Wait, the matrix M = [[1, i], [1, -i]]. The trace of M is 1 + (-i) = 1 - i. The determinant is (1)(-i) - (i)(1) = -i - i = -2i. For Möbius transformations, the trace and determinant relate to the type of transformation. If the trace squared is greater than 4 times the determinant, it's hyperbolic, etc. But maybe I need to find the eigenvalues again. Wait, earlier I found that the eigenvalues are λ₁ and λ₂, which are [1 - i ± sqrt(3)(1 + i)] / 2. Let me compute their product. The product of eigenvalues is determinant of the matrix, which is -2i. So λ₁ * λ₂ = -2i. Also, the trace is λ₁ + λ₂ = 1 - i.

But how does this help me? Maybe if I can find the eigenvalues, then the matrix can be diagonalized, and then its powers can be computed. Let me try that. Suppose I can diagonalize M. Let me denote the eigenvalues as λ₁ and λ₂. Then, M^n = P D^n P^{-1}, where D is the diagonal matrix of eigenvalues. Then, the Möbius transformation corresponding to M^n would be the transformation that maps z to (a z + b)/(c z + d), where the matrix is M^n. However, I need to relate this to the iteration of F(z). Since each iteration corresponds to multiplying by M, then after n iterations, the transformation would be M^n. Therefore, if I can find M^n, then I can find the nth iterate of F(z).

But how do I compute M^n? Well, if I can diagonalize M, then it's straightforward. Let me try to find eigenvectors. Let me take the eigenvalues λ₁ and λ₂. Let me first compute λ₁ and λ₂ numerically to see if they are complex numbers. Let me compute λ₁:

λ₁ = [1 - i + sqrt(3)(1 + i)] / 2

Let me compute sqrt(3)(1 + i) ≈ 1.732*(1 + i) ≈ 1.732 + 1.732i

Then, adding 1 - i: 1 - i + 1.732 + 1.732i ≈ (1 + 1.732) + (-i + 1.732i) ≈ 2.732 + 0.732i

Divide by 2: ≈ 1.366 + 0.366i

Similarly, λ₂ = [1 - i - sqrt(3)(1 + i)] / 2 ≈ [1 - i - 1.732 - 1.732i]/2 ≈ (-0.732 - 2.732i)/2 ≈ -0.366 - 1.366i

So the eigenvalues are complex numbers. Therefore, the matrix M is diagonalizable with complex eigenvalues. Therefore, M^n can be expressed in terms of λ₁^n and λ₂^n. However, since the eigenvalues are complex, maybe they are conjugates? Let me check:

λ₁ ≈ 1.366 + 0.366i

λ₂ ≈ -0.366 - 1.366i

Wait, not sure if they are conjugates. Let me check if λ₂ is the conjugate of λ₁. The conjugate of λ₁ is 1.366 - 0.366i, which is not equal to λ₂. So they are not conjugates. Therefore, the matrix is not diagonalizable over real numbers, but over complex numbers. However, since we have two distinct eigenvalues, it is diagonalizable.

But maybe there's another approach. Let me recall that for Möbius transformations, if we have a transformation with two fixed points, then it can be conjugated to a translation. Let me try again with the fixed points α and β. Let me define w(z) = (z - α)/(z - β). Then, as before, w(F(z)) = w(z) + c. Let me try to compute c.

Let me compute w(F(z)) - w(z) = c. Let me compute this difference:

w(F(z)) - w(z) = [ (F(z) - α)/(F(z) - β) ] - [ (z - α)/(z - β) ] = c

Let me compute this expression. Let me denote F(z) = (z + i)/(z - i). Let me compute numerator and denominator:

First, compute F(z) - α = [(z + i)/(z - i) - α] = [ (z + i) - α(z - i) ] / (z - i) = [ z + i - α z + α i ] / (z - i) = [ (1 - α) z + i(1 + α) ] / (z - i)

Similarly, F(z) - β = [ (1 - β) z + i(1 + β) ] / (z - i)

Therefore, w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

Then, w(F(z)) - w(z) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] - [ (z - α)/(z - β) ] = c

This seems complicated, but maybe I can plug in z = α? Wait, but z = α is a fixed point, so F(α) = α, so w(F(α)) = w(α) = 0, and w(α) = 0, so 0 - 0 = c? That would imply c = 0, which can't be right. Wait, maybe not. Wait, if I plug z = α into the expression, then the first term is [ (1 - α) α + i(1 + α) ] / [ (1 - β) α + i(1 + β) ] and the second term is [ (α - α)/(α - β) ] = 0. Therefore, the difference is [ (1 - α) α + i(1 + α) ] / [ (1 - β) α + i(1 + β) ] = c. But since α is a fixed point, maybe we can use the fixed point equation?

Alternatively, maybe I can use the fact that F(z) = (z + i)/(z - i). Let me try to compute c by choosing a specific z. For example, take z = 0. Let me compute w(F(0)) - w(0) = c.

First, compute F(0) = (0 + i)/(0 - i) = i / (-i) = -1. Then, w(F(0)) = w(-1) = (-1 - α)/(-1 - β). Also, w(0) = (0 - α)/(0 - β) = (-α)/(-β) = α/β. Therefore, c = [ (-1 - α)/(-1 - β) ] - [ α/β ].

This seems messy, but maybe I can compute it using the expressions for α and β. Recall that α and β are fixed points of F(z), so they satisfy F(α) = α and F(β) = β. Let me recall that α and β satisfy the equation z = (z + i)/(z - i). Which led to the quadratic equation z² - (1 + i) z - i = 0. Therefore, α² - (1 + i) α - i = 0 and similarly for β.

Therefore, α² = (1 + i) α + i. Similarly for β².

Let me try to compute c using z = 0. Let me compute:

c = [ (-1 - α)/(-1 - β) ] - [ α/β ]

Let me compute numerator and denominator:

First term: (-1 - α)/(-1 - β) = (1 + α)/(1 + β)

Second term: α/β

Therefore, c = (1 + α)/(1 + β) - α/β

Let me combine these terms:

= [ (1 + α)β - α(1 + β) ] / [ β(1 + β) ]

Compute numerator:

(1 + α)β - α(1 + β) = β + α β - α - α β = β - α

Therefore, c = (β - α)/[ β(1 + β) ]

Hmm, interesting. So c = (β - α)/[ β(1 + β) ]

But I need to compute this. Let me recall that α and β are roots of the equation z² - (1 + i) z - i = 0. Therefore, from Vieta's formula:

α + β = 1 + i

α β = -i

Therefore, β - α can be found as follows. Let me compute (β - α)^2 = (α + β)^2 - 4 α β = (1 + i)^2 - 4*(-i) = (1 + 2i -1) + 4i = 2i + 4i = 6i. Therefore, β - α = sqrt(6i) or -sqrt(6i). Earlier, we saw that sqrt(6i) = sqrt(3)(1 + i). Therefore, β - α = sqrt(3)(1 + i) or -sqrt(3)(1 + i). Wait, but which one is it? Since α and β are the roots, depending on which is which. Let me check with the approximate values earlier. Earlier, α ≈ 1.366 + 1.366i and β ≈ -0.366 - 0.366i. Therefore, β - α ≈ (-0.366 - 0.366i) - (1.366 + 1.366i) ≈ -1.732 - 1.732i. Which is -sqrt(3)(1 + i). Therefore, β - α = -sqrt(3)(1 + i). Therefore, c = [ -sqrt(3)(1 + i) ] / [ β(1 + β) ]

Now, need to compute β(1 + β). Let me compute β(1 + β) = β + β². From the equation β² = (1 + i) β + i. Therefore, β + β² = β + (1 + i) β + i = (1 + i + 1) β + i = (2 + i) β + i. Wait, let me check:

Wait, β² = (1 + i) β + i, so β + β² = β + (1 + i) β + i = [1 + (1 + i)] β + i = (2 + i) β + i. Therefore, β(1 + β) = (2 + i) β + i.

But I need to compute this. Let me compute β. Recall that β is a root of z² - (1 + i) z - i = 0. Let me write β = [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, earlier we had sqrt( (1 + i)^2 + 4i ) = sqrt( -2i + 4i ) = sqrt(2i). Wait, no, earlier when solving for fixed points, we had discriminant D = 6i. Wait, maybe I need to compute β explicitly.

Alternatively, since α + β = 1 + i and α β = -i, then β = (1 + i) - α. Therefore, β(1 + β) = β + β². But β² = (1 + i) β + i, so β + β² = (1 + i) β + i + β = (2 + i) β + i. Therefore, β(1 + β) = (2 + i) β + i. Let me substitute β = [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, but this seems too complicated. Maybe there's another way.

Alternatively, since β satisfies β² = (1 + i) β + i, then β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this expression. Let me denote β(1 + β) = (2 + i) β + i. Let me compute this:

But I need to find β(1 + β). Let me use Vieta's formula. Since α + β = 1 + i and α β = -i. Let me compute β(1 + β) = β + β². From β² = (1 + i) β + i, so β + β² = β + (1 + i) β + i = (1 + i + 1) β + i = (2 + i) β + i. But I need to find this value. Let me think. Maybe I can express β in terms of α? Since β = (1 + i) - α. Therefore:

β(1 + β) = (2 + i) β + i = (2 + i)( (1 + i) - α ) + i. But this seems to bring me back to previous expressions. Maybe I need to compute numerically?

Alternatively, let me use the approximate value of β ≈ -0.366 - 0.366i. Let me compute β(1 + β):

First, compute 1 + β ≈ 1 + (-0.366 - 0.366i) ≈ 0.634 - 0.366i

Then, β(1 + β) ≈ (-0.366 - 0.366i)(0.634 - 0.366i)

Let me compute this:

Multiply the real parts: (-0.366)(0.634) ≈ -0.232

Multiply the cross terms: (-0.366)(-0.366i) + (-0.366i)(0.634) ≈ 0.134i - 0.232i ≈ -0.098i

Multiply the imaginary parts: (-0.366i)(-0.366i) = (0.134)i² = -0.134

So total ≈ -0.232 - 0.134 - 0.098i ≈ -0.366 - 0.098i

But this is approximate. However, I need an exact value. Let me try again with exact expressions.

Let me recall that β satisfies β² = (1 + i) β + i. Therefore, β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this:

(2 + i) β + i = (2 + i) β + i. Let me express β in terms of α. Since α + β = 1 + i, so β = 1 + i - α. Therefore:

(2 + i)(1 + i - α) + i = (2 + i)(1 + i) - (2 + i) α + i

Compute (2 + i)(1 + i) = 2(1) + 2i + i(1) + i(i) = 2 + 2i + i + i² = 2 + 3i -1 = 1 + 3i

Therefore, total expression is 1 + 3i - (2 + i) α + i = 1 + 3i + i - (2 + i) α = 1 + 4i - (2 + i) α

But α is another root, which satisfies α² = (1 + i) α + i. Also, α β = -i. Therefore, maybe I can express α in terms of β? Since α = (1 + i) - β. Therefore:

1 + 4i - (2 + i)(1 + i - β) = 1 + 4i - (2 + i)(1 + i) + (2 + i) β

Compute (2 + i)(1 + i) = 2(1) + 2i + i(1) + i(i) = 2 + 2i + i -1 = 1 + 3i, same as before. Therefore:

1 + 4i - (1 + 3i) + (2 + i) β = (1 + 4i - 1 - 3i) + (2 + i) β = i + (2 + i) β

But this seems to loop back. Maybe I need another approach. Let me recall that α β = -i. Therefore, β = -i / α. Therefore, substituting into β(1 + β):

β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i = (2 + i)(-i / α) + i = [ -i(2 + i) / α ] + i

Compute -i(2 + i) = -2i - i² = -2i + 1 = 1 - 2i. Therefore:

β(1 + β) = (1 - 2i)/α + i

But α is a root of z² - (1 + i) z - i = 0. Therefore, α² = (1 + i) α + i. Let me see if I can express 1/α in terms of α. From α² = (1 + i) α + i, divide both sides by α²:

1 = (1 + i)/α + i / α²

But this seems not helpful. Alternatively, from α² = (1 + i) α + i, rearrange:

α² - (1 + i) α = i

Divide both sides by α:

α - (1 + i) = i / α => α - (1 + i) = i / α => α - (1 + i) - i / α = 0. Not helpful.

Alternatively, maybe I can use the expression for β(1 + β) = (2 + i) β + i and then use β = -i / α:

β(1 + β) = (2 + i)(-i / α) + i = [ -i(2 + i) ] / α + i = [ (1 - 2i) ] / α + i

But from α² = (1 + i) α + i, we can solve for 1/α:

From α² = (1 + i) α + i => α² - (1 + i) α = i => α(α - (1 + i)) = i => α - (1 + i) = i / α => α = (1 + i) + i / α. Not helpful.

Alternatively, maybe I need to accept that this approach is too complicated and try another way. Let me recall that earlier, I had c = (β - α)/[ β(1 + β) ] and β - α = -sqrt(3)(1 + i). Therefore, c = -sqrt(3)(1 + i)/[ β(1 + β) ]

But if I can find β(1 + β), maybe I can relate it to something else. Let me think. From β² = (1 + i) β + i, then β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this expression:

(2 + i) β + i. Let me write β as [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, earlier we had the fixed points as [ (1 + i)(1 ± sqrt(3)) ] / 2. Wait, earlier when solving for fixed points, we had:

z = [ (1 + i)(1 ± sqrt(3)) ] / 2. Wait, but that was for the fixed points of F(z). Wait, but earlier I had that the fixed points satisfy z² - (1 + i) z - i = 0, which has solutions [ (1 + i) ± sqrt( (1 + i)^2 + 4i ) ] / 2. Let me compute (1 + i)^2 + 4i:

(1 + i)^2 = 1 + 2i + i² = 1 + 2i -1 = 2i. Then, adding 4i gives 6i. Therefore, sqrt(6i) = sqrt(3)(1 + i). Therefore, the fixed points are [ (1 + i) ± sqrt(3)(1 + i) ] / 2 = [ (1 + i)(1 ± sqrt(3)) ] / 2. Therefore, β is [ (1 + i)(1 - sqrt(3)) ] / 2. Therefore, β = (1 + i)(1 - sqrt(3))/2. Therefore, I can compute β(1 + β):

First, compute β = (1 + i)(1 - sqrt(3))/2. Let me compute 1 + β:

1 + β = 1 + (1 + i)(1 - sqrt(3))/2 = [ 2 + (1 + i)(1 - sqrt(3)) ] / 2

Let me compute (1 + i)(1 - sqrt(3)) = 1*(1 - sqrt(3)) + i*(1 - sqrt(3)) = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i)

Therefore, 1 + β = [ 2 + (1 - sqrt(3))(1 + i) ] / 2

Therefore, β(1 + β) = [ (1 + i)(1 - sqrt(3))/2 ] * [ 2 + (1 - sqrt(3))(1 + i) ] / 2

Let me compute this:

Let me denote A = (1 + i)(1 - sqrt(3))/2 and B = [ 2 + (1 - sqrt(3))(1 + i) ] / 2. Then β(1 + β) = A * B.

Let me compute A * B:

First, compute (1 - sqrt(3))(1 + i) = (1 - sqrt(3))(1 + i). Let me compute this:

= 1*(1 + i) - sqrt(3)*(1 + i) = (1 + i) - sqrt(3)(1 + i) = (1 - sqrt(3))(1 + i)

Wait, that's the same as before. Let me compute the value:

Let me compute 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me compute this term:

Let me compute (1 - sqrt(3))(1 + i) = (1)(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i). So 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me denote this as C = 2 + D, where D = (1 - sqrt(3))(1 + i).

Therefore, β(1 + β) = A * B = [ (1 + i)(1 - sqrt(3))/2 ] * [ (2 + D)/2 ] = [ (1 + i)(1 - sqrt(3)) * (2 + D) ] / 4

But D = (1 - sqrt(3))(1 + i), so 2 + D = 2 + (1 - sqrt(3))(1 + i). Let me compute this:

Let me compute 2 + (1 - sqrt(3))(1 + i). Let me expand (1 - sqrt(3))(1 + i):

= 1*(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i)

So 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me compute this numerically? Maybe not. Let me think if there's a relation.

Alternatively, note that A = (1 + i)(1 - sqrt(3))/2 and B = [2 + (1 - sqrt(3))(1 + i)] / 2. Let me denote K = (1 - sqrt(3))(1 + i). Then, A = (1 + i)(1 - sqrt(3))/2 = K / 2. And B = [2 + K]/2. Therefore, β(1 + β) = (K / 2) * (2 + K)/2 = K(2 + K)/4.

So β(1 + β) = K(2 + K)/4, where K = (1 - sqrt(3))(1 + i). Let me compute K:

K = (1 - sqrt(3))(1 + i) = (1)(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i). Wait, that's the same as before. Let me compute K numerically:

sqrt(3) ≈ 1.732, so 1 - sqrt(3) ≈ -0.732. Therefore, K ≈ (-0.732)(1 + i) ≈ -0.732 - 0.732i. Then, 2 + K ≈ 2 - 0.732 - 0.732i ≈ 1.268 - 0.732i. Then, K(2 + K) ≈ (-0.732 - 0.732i)(1.268 - 0.732i). Let me compute this:

First, multiply real parts: -0.732 * 1.268 ≈ -0.927

Cross terms: -0.732 * (-0.732i) + (-0.732i) * 1.268 ≈ 0.536i - 0.927i ≈ -0.391i

Imaginary parts: (-0.732i)(-0.732i) = (0.536)i² = -0.536

Total ≈ -0.927 - 0.536 - 0.391i ≈ -1.463 - 0.391i

Then, β(1 + β) ≈ (-1.463 - 0.391i)/4 ≈ -0.366 - 0.098i. Which matches the earlier approximate calculation.

But I need an exact value. Let me try to compute K(2 + K):

K = (1 - sqrt(3))(1 + i)

Let me compute K(2 + K):

= 2K + K²

First, compute 2K = 2(1 - sqrt(3))(1 + i)

Then, compute K² = [(1 - sqrt(3))(1 + i)]²

Let me compute K²:

= (1 - sqrt(3))² (1 + i)²

Compute (1 - sqrt(3))² = 1 - 2 sqrt(3) + 3 = 4 - 2 sqrt(3)

Compute (1 + i)² = 2i

Therefore, K² = (4 - 2 sqrt(3)) * 2i = 2i(4 - 2 sqrt(3)) = 8i - 4 sqrt(3) i

Therefore, K(2 + K) = 2K + K² = 2(1 - sqrt(3))(1 + i) + 8i - 4 sqrt(3) i

Let me compute 2(1 - sqrt(3))(1 + i):

= 2(1 + i - sqrt(3) - sqrt(3)i) = 2(1 - sqrt(3)) + 2i(1 - sqrt(3))

Therefore, K(2 + K) = 2(1 - sqrt(3)) + 2i(1 - sqrt(3)) + 8i - 4 sqrt(3) i

Combine like terms:

Real parts: 2(1 - sqrt(3))

Imaginary parts: 2(1 - sqrt(3))i + 8i - 4 sqrt(3) i

Let me compute the imaginary parts:

Factor out i:

[2(1 - sqrt(3)) + 8 - 4 sqrt(3)] i

Compute inside the brackets:

2(1 - sqrt(3)) + 8 - 4 sqrt(3) = 2 - 2 sqrt(3) + 8 - 4 sqrt(3) = 10 - 6 sqrt(3)

Therefore, K(2 + K) = 2(1 - sqrt(3)) + (10 - 6 sqrt(3))i

Therefore, β(1 + β) = K(2 + K)/4 = [2(1 - sqrt(3)) + (10 - 6 sqrt(3))i]/4 = [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2

Therefore, going back to c = (β - α)/[ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2

Wait, no. Earlier, c = (β - α)/[ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2 ] = [ -2 sqrt(3)(1 + i) ] / [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ]

This seems very complicated. Maybe I need to rationalize the denominator? Let me denote denominator as D = (1 - sqrt(3)) + (5 - 3 sqrt(3))i. Let me compute D:

Let me write D = a + bi, where a = 1 - sqrt(3), b = 5 - 3 sqrt(3). Then, c = [ -2 sqrt(3)(1 + i) ] / (a + bi). Multiply numerator and denominator by the conjugate of the denominator, which is a - bi:

c = [ -2 sqrt(3)(1 + i)(a - bi) ] / [ (a + bi)(a - bi) ] = [ -2 sqrt(3)(1 + i)(a - bi) ] / (a² + b²)

Compute denominator a² + b²:

a = 1 - sqrt(3), so a² = (1 - sqrt(3))² = 1 - 2 sqrt(3) + 3 = 4 - 2 sqrt(3)

b = 5 - 3 sqrt(3), so b² = (5)^2 - 2*5*3 sqrt(3) + (3 sqrt(3))² = 25 - 30 sqrt(3) + 27 = 52 - 30 sqrt(3)

Therefore, a² + b² = (4 - 2 sqrt(3)) + (52 - 30 sqrt(3)) = 56 - 32 sqrt(3)

Now, compute numerator: -2 sqrt(3)(1 + i)(a - bi). Let me compute (1 + i)(a - bi):

= 1*(a - bi) + i*(a - bi) = a - bi + ai - b i² = a - bi + ai + b = (a + b) + i(a - b)

Therefore, numerator becomes -2 sqrt(3) [ (a + b) + i(a - b) ]

Compute a + b = (1 - sqrt(3)) + (5 - 3 sqrt(3)) = 6 - 4 sqrt(3)

Compute a - b = (1 - sqrt(3)) - (5 - 3 sqrt(3)) = 1 - sqrt(3) - 5 + 3 sqrt(3) = -4 + 2 sqrt(3)

Therefore, numerator is -2 sqrt(3) [ (6 - 4 sqrt(3)) + i(-4 + 2 sqrt(3)) ] = -2 sqrt(3)(6 - 4 sqrt(3)) - 2 sqrt(3) i(-4 + 2 sqrt(3))

Let me compute each part:

First term: -2 sqrt(3)(6 - 4 sqrt(3)) = -12 sqrt(3) + 8 * 3 = -12 sqrt(3) + 24

Second term: -2 sqrt(3) * i * (-4 + 2 sqrt(3)) = 2 sqrt(3) i (4 - 2 sqrt(3)) = 8 sqrt(3) i - 4 * 3 i = 8 sqrt(3) i - 12 i

Therefore, numerator is (-12 sqrt(3) + 24) + (8 sqrt(3) - 12)i

Therefore, c = [ (-12 sqrt(3) + 24) + (8 sqrt(3) - 12)i ] / (56 - 32 sqrt(3))

This is getting extremely complicated. I must be missing a simpler approach. Let me think again.

Maybe instead of trying to find c, I can use the fact that if the transformation is conjugate to a translation, then the iteration becomes w_{n} = w_0 + n*c. Therefore, if I can find w_0 = w(z_0) and c, then w_{2002} = w_0 + 2002*c, and then z_{2002} = w^{-1}(w_{2002}).

But since I'm stuck on finding c, maybe I can try to compute w(z_0) and see if there's a pattern?

Let me recall that w(z) = (z - α)/(z - β). Given that z_0 = 1/137 + i, and α and β are the fixed points. Let me compute w(z_0):

w(z_0) = (z_0 - α)/(z_0 - β)

But without knowing α and β explicitly, this is difficult. However, maybe I can relate this to the earlier expressions. Let me recall that α and β are roots of z² - (1 + i) z - i = 0. Let me denote z_0 = 1/137 + i. Let me compute w(z_0):

Let me write w(z_0) = (z_0 - α)/(z_0 - β). Let me note that since α and β are roots of the quadratic equation, then (z - α)(z - β) = z² - (α + β) z + α β = z² - (1 + i) z - i. Therefore, for any z, (z - α)(z - β) = z² - (1 + i) z - i. Therefore, if I can express (z_0 - α)/(z_0 - β) in terms of z_0, maybe?

Let me denote w(z_0) = (z_0 - α)/(z_0 - β). Let me note that (z_0 - α)/(z_0 - β) = [ (z_0 - α) / (z_0 - β) ] = [ (z_0 - α) / (z_0 - β) ].

Alternatively, since (z_0 - α)(z_0 - β) = z_0² - (1 + i) z_0 - i, then (z_0 - α) = [ z_0² - (1 + i) z_0 - i ] / (z_0 - β). Therefore, w(z_0) = [ z_0² - (1 + i) z_0 - i ] / (z_0 - β)^2. Not sure if helpful.

Alternatively, maybe I can use the fact that if w(F(z)) = w(z) + c, then w(z_n) = w(z_0) + n*c. Therefore, if I can find w(z_0) and c, then I can find w(z_{2002}) = w(z_0) + 2002*c, and then invert the transformation to get z_{2002}.

But since I can't find c easily, maybe I can find c by using the fact that the transformation is conjugate to a translation. Let me recall that the transformation F(z) has a certain multiplier, which is the ratio of the eigenvalues or something. Wait, earlier I had the matrix M with eigenvalues λ₁ and λ₂. The trace of M is 1 - i, and the determinant is -2i. For Möbius transformations, the multiplier is the trace squared minus 4 times the determinant divided by something? Wait, no. The multiplier of a Möbius transformation is the derivative at a fixed point. Let me compute the derivative of F(z).

F(z) = (z + i)/(z - i). The derivative F’(z) = [ (1)(z - i) - (z + i)(1) ] / (z - i)^2 = [ z - i - z - i ] / (z - i)^2 = (-2i) / (z - i)^2. At a fixed point α, F’(α) = -2i / (α - i)^2. But since α is a fixed point, F(α) = α, so maybe I can relate this to the eigenvalues?

Alternatively, since the transformation is conjugate to a translation, the multiplier should be 1, since translations have derivative 1. Wait, but if it's conjugate to a translation, then the multiplier should be 1. However, the multiplier of F(z) is the derivative at a fixed point. Let me compute it.

Let me compute F’(z) = -2i / (z - i)^2. At the fixed point α, F’(α) = -2i / (α - i)^2. Let me compute α - i. Since α is a fixed point, α = (z + i)/(z - i) = z. Wait, no, α is a fixed point, so F(α) = α. Let me compute α - i. Let me recall that α satisfies α = (α + i)/(α - i). Let me solve for α - i:

Multiply both sides by α - i: α(α - i) = α + i

Expand: α² - i α = α + i

Rearrange: α² - i α - α - i = 0 => α² - (1 + i) α - i = 0, which is the original equation. Not helpful. Let me compute α - i:

From α = (α + i)/(α - i), rearrange:

α(α - i) = α + i => α² - i α = α + i => α² = (1 + i) α + i. Let me compute α - i:

Let me denote α - i = k. Then, α = k + i. Substitute into α² = (1 + i) α + i:

(k + i)^2 = (1 + i)(k + i) + i

Expand left side: k² + 2i k + i² = k² + 2i k - 1

Right side: (1 + i)k + (1 + i)i + i = (1 + i)k + i + i² + i = (1 + i)k + i - 1 + i = (1 + i)k + 2i - 1

Set equal:

k² + 2i k - 1 = (1 + i)k + 2i - 1

Subtract -1 from both sides:

k² + 2i k = (1 + i)k + 2i

Bring all terms to left:

k² + 2i k - (1 + i)k - 2i = 0 => k² + [2i - 1 - i]k - 2i = 0 => k² + (i - 1)k - 2i = 0

This is the same equation as before for λ! Wait, earlier when solving for eigenvalues, we had λ² + (i - 1)λ - 2i = 0. So this suggests that k = λ, the eigenvalues. Therefore, α - i = λ, so F’(α) = -2i / (α - i)^2 = -2i / λ². But since λ is an eigenvalue of the matrix M, and the trace of M is 1 - i, and the determinant is -2i, then for the eigenvalues λ₁ and λ₂, we have λ₁ + λ₂ = 1 - i and λ₁ λ₂ = -2i.

But if F’(α) = -2i / λ², and since the multiplier of the transformation is related to the eigenvalues, but I'm not sure. However, if the transformation is conjugate to a translation, then the multiplier should be 1, but here it's not necessarily. Wait, if it's conjugate to a translation, then the multiplier should be 1, but the multiplier here is F’(α) = -2i / λ². Let me compute this for λ₁.

Let me take λ₁ = [1 - i + sqrt(3)(1 + i)] / 2. Let me compute λ₁². This seems complicated, but maybe there's a relation. Since λ₁ and λ₂ are roots of λ² + (i - 1)λ - 2i = 0, then λ₁² = -(i - 1)λ₁ + 2i. Therefore, F’(α) = -2i / λ₁² = -2i / [ -(i - 1)λ₁ + 2i ] = -2i / [ (1 - i)λ₁ + 2i ].

This seems too complex. Maybe I need to accept that this approach is too difficult and try another way.

Let me go back to the original problem. We have z_0 = 1/137 + i, and we need to find z_{2002} under the iteration z_n = F(z_{n-1}).

Maybe I can compute the first few terms and see if there's a pattern or periodicity.

Let me compute z_0 = 1/137 + i.

Compute z_1 = F(z_0) = (z_0 + i)/(z_0 - i).

Let me compute numerator and denominator:

Numerator: z_0 + i = (1/137 + i) + i = 1/137 + 2i

Denominator: z_0 - i = (1/137 + i) - i = 1/137

Therefore, z_1 = (1/137 + 2i) / (1/137) = 1 + 2i * 137 = 1 + 274i

Wait, that's a big jump! So z_1 = 1 + 274i.

Now compute z_2 = F(z_1) = (z_1 + i)/(z_1 - i) = (1 + 274i + i)/(1 + 274i - i) = (1 + 275i)/(1 + 273i)

Let me rationalize this fraction. Multiply numerator and denominator by the conjugate of the denominator, which is 1 - 273i:

Numerator: (1 + 275i)(1 - 273i) = 1*1 + 1*(-273i) + 275i*1 + 275i*(-273i) = 1 - 273i + 275i - 275*273 i²

Simplify:

= 1 + 2i - 275*273*(-1) = 1 + 2i + 275*273

Compute 275*273: Let me compute 275*273. Note that 275 = 25*11, 273 = 3*7*13. Let me compute 275*273:

= (270 + 5)(270 + 3) = 270² + 270*3 + 5*270 + 5*3 = 72900 + 810 + 1350 + 15 = 72900 + 810 = 73710 + 1350 = 75060 + 15 = 75075. Wait, but this is for (270 + 5)(270 + 3). Wait, but 275*273 = (270 + 5)(270 + 3) = 270² + 270*3 + 5*270 + 5*3 = 72900 + 810 + 1350 + 15 = 72900 + 2175 = 75075. Yes.

So numerator = 1 + 2i + 75075 = 75076 + 2i

Denominator: (1 + 273i)(1 - 273i) = 1² - (273i)^2 = 1 - 273² * (-1) = 1 + 273²

Compute 273²: 273*273. Let me compute 270² + 2*270*3 + 3² = 72900 + 1620 + 9 = 74529. Therefore, denominator = 1 + 74529 = 74530

Therefore, z_2 = (75076 + 2i)/74530. Let me simplify this fraction. Let me divide numerator and denominator by 2:

Numerator: 75076 / 2 = 37538, 2i / 2 = i

Denominator: 74530 / 2 = 37265

So z_2 = (37538 + i)/37265. Let me see if this can be simplified further. Let me check if 37538 and 37265 have a common factor. Let me compute GCD(37538, 37265).

Compute 37538 - 37265 = 273. Now GCD(37265, 273).

Divide 37265 by 273: 273*136 = 273*(100 + 30 + 6) = 27300 + 8190 + 1638 = 27300 + 9828 = 37128. 37265 - 37128 = 137. Now GCD(273, 137).

273 ÷ 137 = 2 with remainder 273 - 2*137 = 273 - 274 = -1? Wait, 2*137 = 274, which is greater than 273. So 1*137 = 137, remainder 273 - 137 = 136. Now GCD(137, 136). Which is 1. Therefore, GCD is 1. So the fraction cannot be simplified further. Therefore, z_2 = (37538 + i)/37265.

This seems messy. Let me compute z_2 numerically to see if there's a pattern. Let me approximate:

Numerator: 37538 + i ≈ 37538 + 0i + i ≈ 37538 + i

Denominator: 37265

So z_2 ≈ (37538/37265) + (1/37265)i ≈ 1.0073 + 0.000027i. So approximately 1.0073 + 0.000027i. Very close to 1.

Wait, this is interesting. Let me check my calculation for z_2. Wait, z_1 = 1 + 274i. Then z_2 = (1 + 274i + i)/(1 + 274i - i) = (1 + 275i)/(1 + 273i). When I rationalized, I got numerator 75076 + 2i and denominator 74530. Let me check that again.

Wait, (1 + 275i)(1 - 273i) = 1*1 + 1*(-273i) + 275i*1 + 275i*(-273i) = 1 - 273i + 275i - 275*273*i². Since i² = -1, this becomes 1 + 2i + 275*273. Yes, and 275*273 = 75075, so total numerator is 1 + 2i + 75075 = 75076 + 2i. Denominator is 1 + 273² = 1 + 74529 = 74530. Correct. So z_2 ≈ 75076/74530 + (2/74530)i ≈ 1.0073 + 0.000027i. So very close to 1.

Now compute z_3 = F(z_2) = (z_2 + i)/(z_2 - i). Since z_2 is approximately 1 + 0.000027i, let me approximate z_2 ≈ 1 + εi, where ε is very small. Then:

z_3 ≈ (1 + εi + i)/(1 + εi - i) = (1 + i(ε + 1))/(1 - i(1 - ε)) ≈ [1 + i(1 + ε)] / [1 - i(1 - ε)]

If ε is very small, we can approximate this as [1 + i]/[1 - i] = (1 + i)^2 / (1 + 1) = (1 + 2i -1)/2 = 2i / 2 = i. So z_3 ≈ i. Let me check with exact values.

Wait, let me compute z_2 = (37538 + i)/37265. Let me compute z_2 + i = (37538 + i)/37265 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265

Therefore, z_3 = [37538 + 37266i] / [37538 - 37264i]

Let me rationalize this by multiplying numerator and denominator by the conjugate of the denominator, which is 37538 + 37264i:

Numerator: (37538 + 37266i)(37538 + 37264i)

Denominator: (37538)^2 + (37264)^2

This seems very complex, but let me note that the denominator is a large number, and the numerator might simplify. Let me see if there's a pattern.

Alternatively, since z_2 is very close to 1, let me consider that z_2 = 1 + ε, where ε is small. Then, z_3 = (1 + ε + i)/(1 + ε - i). Let me compute this:

= [ (1 + i) + ε ] / [ (1 - i) + ε ] ≈ [ (1 + i) + ε ] / (1 - i) * [1 - ε/(1 - i) + ... ] using the approximation 1/(1 + x) ≈ 1 - x for small x.

But this might not help. Alternatively, let me compute z_3 exactly. Let me note that z_2 = (37538 + i)/37265. Let me compute z_2 + i = (37538 + i)/37265 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265

Now, let me compute the numerator and denominator:

Numerator: 37538 + 37266i

Denominator: 37538 - 37264i

Let me note that 37538 = 37265 + 273. Let me write this as:

Numerator: (37265 + 273) + 37266i

Denominator: (37265 + 273) - 37264i

Let me factor out 37265 from numerator and denominator:

Numerator: 37265 + 273 + 37266i = 37265(1) + 273 + 37266i

Denominator: 37265 + 273 - 37264i = 37265(1) + 273 - 37264i

This doesn't seem helpful. Let me try to compute the ratio:

Let me denote A = 37538, B = 37266, C = 37538, D = 37264. Wait, no, numerator is A + Bi, denominator is A - Di, where A = 37538, B = 37266, D = 37264.

Let me compute the ratio (A + Bi)/(A - Di). Let me multiply numerator and denominator by A + Di:

Numerator: (A + Bi)(A + Di) = A² + ADi + ABi + BDi² = A² + (AD + AB)i - BD

Denominator: (A - Di)(A + Di) = A² + D²

So z_3 = [A² - BD + (AD + AB)i] / (A² + D²)

Let me compute A, B, D:

A = 37538

B = 37266

D = 37264

Compute A² - BD: Let me compute BD = 37266 * 37264. Note that 37266 * 37264 = (37265 + 1)(37265 - 1) = 37265² - 1

Similarly, A = 37538 = 37265 + 273

So A² = (37265 + 273)^2 = 37265² + 2*37265*273 + 273²

Therefore, A² - BD = 37265² + 2*37265*273 + 273² - (37265² - 1) = 2*37265*273 + 273² + 1

Let me compute this:

= 2*37265*273 + 273² + 1

= 273*(2*37265 + 273) + 1

Compute 2*37265 = 74530, so 74530 + 273 = 74803

Thus, A² - BD = 273*74803 + 1

This is a large number, but let me note that this is the real part of the numerator. The imaginary part is (AD + AB)i = A(D + B)i. Let me compute D + B = 37264 + 37266 = 74530. So the imaginary part is A * 74530 * i = 37538 * 74530 * i.

Therefore, z_3 = [273*74803 + 1 + 37538*74530 i] / (A² + D²)

This is extremely complex, but let me note that the denominator is A² + D² = (37538)^2 + (37264)^2. This is also a large number. However, given that z_2 was very close to 1, and z_3 is approximately i, maybe there's a pattern here. Let me check if z_3 is i.

If z_3 = i, then let me check if (z_2 + i)/(z_2 - i) = i. Let me solve for z_2:

(z_2 + i)/(z_2 - i) = i => z_2 + i = i(z_2 - i) => z_2 + i = i z_2 - i² => z_2 + i = i z_2 + 1 => z_2 - i z_2 = 1 - i => z_2(1 - i) = 1 - i => z_2 = 1.

But z_2 is approximately 1.0073, not exactly 1. So z_3 is not exactly i, but very close to it. Let me compute the exact value of z_3. Given that z_2 is very close to 1, let me assume that z_2 = 1 + ε, where ε is small. Then:

z_3 = (1 + ε + i)/(1 + ε - i) = [ (1 + i) + ε ] / [ (1 - i) + ε ] ≈ [ (1 + i) + ε ] / (1 - i) * [1 - ε/(1 - i) + ... ]

But this is getting too approximate. Alternatively, let me compute z_3 using the earlier expression:

z_3 = [A² - BD + (AD + AB)i] / (A² + D²). Let me compute A² - BD = 273*74803 + 1. Let me compute 273*74803. Let me note that 74803 = 74530 + 273. So 273*74803 = 273*74530 + 273^2.

But this is too tedious. Let me instead notice that if z_2 is very close to 1, then z_3 is very close to i, and then z_4 would be F(i) = (i + i)/(i - i) = 2i / 0, which is undefined. Wait, but z_2 is not exactly 1, so z_3 is not exactly i, but very close. Let me compute z_3 more accurately.

Given that z_2 ≈ 1 + ε, where ε = (37538 - 37265)/37265 + i*(1/37265) ≈ 273/37265 + i/37265 ≈ 0.0073 + 0.000027i. So ε ≈ 0.0073 + 0.000027i. Then, z_3 = (1 + ε + i)/(1 + ε - i) ≈ (1 + i + ε)/(1 - i + ε). Let me write this as [(1 + i) + ε]/[(1 - i) + ε]. Let me expand this using the approximation 1/(1 - i + ε) ≈ 1/(1 - i) * 1/(1 + ε/(1 - i)) ≈ 1/(1 - i) * [1 - ε/(1 - i) + ...].

So z_3 ≈ [(1 + i) + ε] * [1/(1 - i) - ε/(1 - i)^2 + ...]. Let me compute 1/(1 - i) = (1 + i)/2. So:

z_3 ≈ (1 + i) * (1 + i)/2 - ε * (1 + i)/2 * (1/(1 - i)^2) + ...

Wait, this is getting too complex. Let me compute the first term: (1 + i) * (1 + i)/2 = (1 + 2i -1)/2 = 2i/2 = i. The second term is -ε * (1 + i)/2 * [1/(1 - i)^2]. Let me compute 1/(1 - i)^2 = (1 + i)^2 / (1 + 1)^2 = (2i)/4 = i/2. Therefore, the second term is -ε * (1 + i)/2 * i/2 = -ε * (1 + i)i / 4 = -ε * (i - 1)/4. Since ε is small, this term is negligible. Therefore, z_3 ≈ i - negligible. So z_3 is approximately i.

If z_3 is approximately i, then z_4 = F(z_3) = (i + i)/(i - i) = 2i / 0, which is undefined. But since z_3 is not exactly i, let me compute it more precisely. Let me assume z_3 = i + δ, where δ is very small. Then, z_4 = F(z_3) = (z_3 + i)/(z_3 - i) = (i + δ + i)/(i + δ - i) = (2i + δ)/δ = 2i/δ + 1. If δ is very small, this would be a large number. But this suggests that the sequence might be approaching i, and then diverging. However, this seems inconsistent with the earlier steps.

Alternatively, maybe there's a cycle. Let me check if applying F multiple times brings us back to the original value. For example, if we start from z_0 = 1/137 + i, then z_1 = 1 + 274i, z_2 ≈ 1.0073 + 0.000027i, z_3 ≈ i, and then z_4 would be undefined? This seems problematic. Maybe there's a mistake in my calculations?

Wait, let me check z_3 again. If z_2 is very close to 1, then z_3 = (z_2 + i)/(z_2 - i) ≈ (1 + i)/(1 - i) = i. So z_3 ≈ i. Then z_4 = F(z_3) = (i + i)/(i - i) = 2i / 0, which is undefined. But since z_3 is not exactly i, let me compute it more precisely. Let me take z_2 = (37538 + i)/37265. Let me compute z_2 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265. Let me compute this as a complex number:

Real part: 37538 / 37265 ≈ 1.0073

Imaginary part: -37264 / 37265 ≈ -0.99997

So z_2 - i ≈ 1.0073 - 0.99997i ≈ 1.0073 - i. Wait, no, the imaginary part is -37264/37265 ≈ -1 + 1/37265 ≈ -1 + 0.000027. So z_2 - i ≈ 1.0073 - i + 0.000027i ≈ 1.0073 - i(1 - 0.000027). So approximately 1.0073 - i.

Then, z_2 + i ≈ 1.0073 + i + i = 1.0073 + 2i. Therefore, z_3 ≈ (1.0073 + 2i) / (1.0073 - i). Let me compute this:

Multiply numerator and denominator by the conjugate of the denominator, which is 1.0073 + i:

Numerator: (1.0073 + 2i)(1.0073 + i) ≈ (1.0073)^2 + 1.0073i + 2i*1.0073 + 2i^2

≈ 1.0146 + 1.0073i + 2.0146i - 2

≈ (1.0146 - 2) + (1.0073 + 2.0146)i

≈ -0.9854 + 3.0219i

Denominator: (1.0073)^2 + (1)^2 ≈ 1.0146 + 1 = 2.0146

So z_3 ≈ (-0.9854 + 3.0219i)/2.0146 ≈ -0.489 + 1.500i

Wait, this is different from my previous approximation. So z_3 ≈ -0.489 + 1.5i. Let me check if this is close to i? No, it's not. So my previous assumption was wrong. Let me compute this more accurately.

Wait, let me compute the exact value of z_3. Given that z_2 = (37538 + i)/37265, let me compute z_2 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 - 37264i)/37265

Let me compute the ratio:

Numerator: 37538 + 37266i

Denominator: 37538 - 37264i

Let me divide numerator and denominator by 37265:

Numerator: (37538/37265) + (37266/37265)i ≈ 1.0073 + 1.0000i

Denominator: (37538/37265) - (37264/37265)i ≈ 1.0073 - 0.99997i

So z_3 ≈ (1.0073 + i) / (1.0073 - i). Let me compute this:

Multiply numerator and denominator by (1.0073 + i):

Numerator: (1.0073 + i)^2 = 1.0073² + 2*1.0073*i + i² ≈ 1.0146 + 2.0146i - 1 = 0.0146 + 2.0146i

Denominator: (1.0073)^2 + 1 ≈ 1.0146 + 1 = 2.0146

So z_3 ≈ (0.0146 + 2.0146i)/2.0146 ≈ 0.00725 + i

So z_3 ≈ 0.00725 + i. This is very close to i, with a small real part. Let me call this z_3 ≈ i + ε, where ε = 0.00725. Then, z_4 = F(z_3) = (z_3 + i)/(z_3 - i) = (i + ε + i)/(i + ε - i) = (2i + ε)/ε = 2i/ε + 1. Since ε is small, this is a large number. Let me compute it:

z_4 ≈ 2i / 0.00725 + 1 ≈ 276i + 1. So z_4 ≈ 1 + 276i. Wait, this is similar to z_1 = 1 + 274i. Let me check if there's a pattern here.

Wait, z_1 = 1 + 274i, z_4 ≈ 1 + 276i. Let me compute z_5 = F(z_4) = (z_4 + i)/(z_4 - i) = (1 + 276i + i)/(1 + 276i - i) = (1 + 277i)/(1 + 275i). This is similar to z_2, which was (1 + 275i)/(1 + 273i). So it seems like there's a pattern where each time, the imaginary part increases by 2. Let me check:

z_1 = 1 + 274i

z_4 ≈ 1 + 276i

If this pattern continues, then z_7 would be 1 + 278i, and so on. If this is the case, then the sequence is periodic with period 3? Let me check.

Wait, from z_0 to z_1: +274i

z_1 to z_2: z_2 ≈ 1 + 0.00725i

z_2 to z_3: z_3 ≈ 0.00725 + i

z_3 to z_4: z_4 ≈ 1 + 276i

z_4 to z_5: similar to z_1 to z_2, leading to z_5 ≈ 1 + 0.00725i

Wait, this suggests that after three iterations, the sequence returns to a value close to z_2, and then repeats. If this is the case, then the period is 3. Let me check with exact calculations.

Let me assume that applying F three times brings us back to a value close to the original. Let me see:

If z_1 = 1 + 274i, then z_2 = (1 + 275i)/(1 + 273i), z_3 = (z_2 + i)/(z_2 - i) ≈ 0.00725 + i, and z_4 = F(z_3) ≈ 1 + 276i. Then, z_5 = F(z_4) = (1 + 276i + i)/(1 + 276i - i) = (1 + 277i)/(1 + 275i), which is similar to z_2, but with 277i instead of 275i. So it seems like each time, the imaginary part increases by 2. If this pattern continues, then after three iterations, the imaginary part increases by 2*3 = 6? Wait, from z_1 to z_4, the imaginary part increased by 2 (from 274i to 276i). So each iteration increases the imaginary part by 2? No, from z_1 to z_2, it decreased, then from z_2 to z_3, it increased, and from z_3 to z_4, it increased again. It's not clear.

Alternatively, let me notice that z_1 = 1 + 274i, z_4 ≈ 1 + 276i, which is 2 more. If this pattern continues, then z_7 would be 1 + 278i, and so on. If this is the case, then the sequence is not periodic, but the imaginary part increases by 2 every 3 steps. However, this seems unlikely.

Alternatively, let me consider that the transformation F(z) = (z + i)/(z - i) has a property that if we apply it three times, it might return to the original value or something. Let me try to compute F^3(z).

Let me recall that earlier, I tried to find the eigenvalues and the matrix, but it was too complex. Let me try to find a pattern with the first few terms.

We have:

z_0 = 1/137 + i

z_1 = 1 + 274i

z_2 = (1 + 275i)/(1 + 273i) ≈ 1.0073 + 0.000027i

z_3 ≈ 0.00725 + i

z_4 ≈ 1 + 276i

z_5 = (1 + 277i)/(1 + 275i) ≈ similar to z_2

z_6 ≈ 0.00725 + i

z_7 ≈ 1 + 278i

So it seems like every 3 steps, the imaginary part increases by 2. If this pattern continues, then the sequence is periodic with period 3, but with the imaginary part increasing. However, this seems contradictory. Alternatively, maybe there's a cycle where after three iterations, the value returns to a form similar to the original, but with a different constant.

Alternatively, let me consider that the transformation F(z) = (z + i)/(z - i) can be related to a rotation in the complex plane. Let me recall that if I write z = x + yi, then F(z) = (x + yi + i)/(x + yi - i) = (x + (y + 1)i)/(x + (y - 1)i). This resembles a Möbius transformation that might be a rotation or something else.

Alternatively, let me consider the transformation in terms of the imaginary part. Let me set w = z - i, then z = w + i. Substitute into F(z):

F(z) = (w + i + i)/(w + i - i) = (w + 2i)/w = 1 + 2i/w. So F(z) = 1 + 2i/w, where w = z - i.

Let me define w_n = z_n - i. Then, the recurrence becomes:

w_n = z_n - i = F(z_{n-1}) - i = [(z_{n-1} + i)/(z_{n-1} - i)] - i = [ (z_{n-1} + i) - i(z_{n-1} - i) ] / (z_{n-1} - i) = [ z_{n-1} + i - i z_{n-1} + i² ] / (z_{n-1} - i) = [ z_{n-1}(1 - i) + i - 1 ] / (z_{n-1} - i)

But since w_{n-1} = z_{n-1} - i, then z_{n-1} = w_{n-1} + i. Substitute this into the expression:

Numerator: (w_{n-1} + i)(1 - i) + i - 1

Let me compute this:

= w_{n-1}(1 - i) + i(1 - i) + i - 1

= w_{n-1}(1 - i) + i - i² + i - 1

= w_{n-1}(1 - i) + i + 1 + i - 1

= w_{n-1}(1 - i) + 2i

Therefore, w_n = [ w_{n-1}(1 - i) + 2i ] / (w_{n-1} + i - i) = [ w_{n-1}(1 - i) + 2i ] / w_{n-1} = (1 - i) + 2i / w_{n-1}

So the recurrence for w_n is:

w_n = (1 - i) + 2i / w_{n-1}

This is a recurrence relation for w_n. Let me write it as:

w_n = (1 - i) + 2i / w_{n-1}

This seems more manageable. Let me try to compute the first few terms.

Given z_0 = 1/137 + i, so w_0 = z_0 - i = 1/137.

Compute w_1 = (1 - i) + 2i / w_0 = (1 - i) + 2i / (1/137) = (1 - i) + 274i = 1 + 273i

Wait, earlier I had z_1 = 1 + 274i, so w_1 = z_1 - i = 1 + 274i - i = 1 + 273i. Correct.

Now compute w_2 = (1 - i) + 2i / w_1 = (1 - i) + 2i / (1 + 273i). Let me compute 2i / (1 + 273i). Multiply numerator and denominator by (1 - 273i):

2i(1 - 273i) / (1 + 273i)(1 - 273i) = [2i - 546i²] / (1 + 273²) = [2i + 546] / (1 + 74529) = (546 + 2i)/74530

Therefore, w_2 = (1 - i) + (546 + 2i)/74530 = [ (1 - i) * 74530 + 546 + 2i ] / 74530

Compute numerator:

1 * 74530 = 74530

- i * 74530 = -74530i

+ 546 + 2i

Total: 74530 + 546 + (-74530i + 2i) = 75076 - 74528i

Therefore, w_2 = (75076 - 74528i)/74530 = (75076/74530) - (74528/74530)i ≈ 1.0073 - 0.99997i, which matches our earlier approximation for z_2 - i.

Now compute w_3 = (1 - i) + 2i / w_2. Let me compute 2i / w_2 = 2i / [(75076 - 74528i)/74530] = 2i * 74530 / (75076 - 74528i) = 149060i / (75076 - 74528i). Let me rationalize this by multiplying numerator and denominator by (75076 + 74528i):

Numerator: 149060i * (75076 + 74528i) = 149060i * 75076 + 149060i * 74528i

Denominator: (75076)^2 + (74528)^2

This is very complex, but let me note that this is similar to previous calculations. However, let me notice that this is getting too cumbersome. Let me instead look for a pattern in the w_n sequence.

We have:

w_0 = 1/137

w_1 = 1 + 273i

w_2 = (75076 - 74528i)/74530 ≈ 1.0073 - 0.99997i

w_3 = (1 - i) + 2i / w_2

Let me assume that after some steps, the sequence enters a cycle. Let me try to compute w_3:

Let me denote w_2 = a - bi, where a = 75076/74530 ≈ 1.0073, b = 74528/74530 ≈ 0.99997. Then, 2i / w_2 = 2i / (a - bi) = 2i(a + bi)/ (a² + b²) = 2i(a + bi)/ (a² + b²) = 2i a / (a² + b²) - 2b / (a² + b²)

Therefore, w_3 = (1 - i) + 2i a / (a² + b²) - 2b / (a² + b²)

Let me compute a² + b². Since w_2 = (75076 - 74528i)/74530, then a² + b² = (75076² + 74528²)/74530². Let me compute this:

But this is equal to |w_2|² = |z_2 - i|². Since z_2 is close to 1, this is approximately |1 - i|² = 2. But let me compute it exactly:

From earlier, when we computed z_2 = (37538 + i)/37265, then w_2 = z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265. Therefore, |w_2|² = (37538² + 37264²)/37265². Let me compute 37538² + 37264². Note that 37538 = 37265 + 273, and 37264 = 37265 - 1. Let me compute:

(37265 + 273)^2 + (37265 - 1)^2 = 37265² + 2*37265*273 + 273² + 37265² - 2*37265*1 + 1 = 2*37265² + 2*37265*(273 - 1) + 273² + 1

This is complex, but let me note that this is equal to 2*37265² + 2*37265*272 + 273² + 1. However, this is not helpful. Let me instead note that this is equal to |w_2|² = |z_2 - i|². Since z_2 is close to 1, this is approximately |1 - i|² = 2. Let me assume that a² + b² ≈ 2. Then, w_3 ≈ (1 - i) + 2i a / 2 - 2b / 2 = (1 - i) + i a - b.

With a ≈ 1.0073 and b ≈ 0.99997, this becomes:

(1 - i) + i*1.0073 - 0.99997 ≈ (1 - 0.99997) + (-i + 1.0073i) ≈ 0.00003 + 0.0073i. So w_3 ≈ 0.00003 + 0.0073i. This is very small, close to 0.

Then, w_4 = (1 - i) + 2i / w_3. Since w_3 is very small, 2i / w_3 is very large. Let me approximate w_3 ≈ 0.0073i. Then, 2i / w_3 ≈ 2i / (0.0073i) = 2 / 0.0073 ≈ 274. So w_4 ≈ (1 - i) + 274 ≈ 275 - i.

Then, w_5 = (1 - i) + 2i / w_4 = (1 - i) + 2i / (275 - i). Let me compute 2i / (275 - i) = 2i(275 + i) / (275² + 1) ≈ 2i*275 / 275² = 2i / 275 ≈ 0.0073i. So w_5 ≈ (1 - i) + 0.0073i ≈ 1 - 0.9927i.

Then, w_6 = (1 - i) + 2i / w_5 ≈ (1 - i) + 2i / (1 - 0.9927i). Let me compute 2i / (1 - 0.9927i) ≈ 2i(1 + 0.9927i) / (1 + 0.9927²) ≈ 2i + 1.9854i² / 1.9854 ≈ 2i - 1.9854 / 1.9854 ≈ 2i - 1. So w_6 ≈ (1 - i) + 2i - 1 = (1 - 1) + (-i + 2i) = i.

Then, w_7 = (1 - i) + 2i / w_6 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i.

Then, w_8 = (1 - i) + 2i / w_7 = (1 - i) + 2i / (3 - i) = (1 - i) + 2i(3 + i)/ (9 + 1) = (1 - i) + (6i + 2i²)/10 = (1 - i) + (6i - 2)/10 = (1 - 2/10) + (-i + 6i/10) = 0.8 - 0.4i.

Then, w_9 = (1 - i) + 2i / w_8 = (1 - i) + 2i / (0.8 - 0.4i). Let me compute 2i / (0.8 - 0.4i) = 2i(0.8 + 0.4i) / (0.64 + 0.16) = (1.6i + 0.8i²) / 0.8 = (1.6i - 0.8)/0.8 = -1 + 2i. So w_9 ≈ (1 - i) + (-1 + 2i) = 0 + i.

Then, w_10 = (1 - i) + 2i / w_9 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i, which is the same as w_7. So it seems like from w_6 = i, w_7 = 3 - i, w_8 = 0.8 - 0.4i, w_9 = i, w_10 = 3 - i, and so on. This suggests a cycle of length 3 starting from w_6: i → 3 - i → 0.8 - 0.4i → i → ...

Wait, let me check:

w_6 = i

w_7 = 3 - i

w_8 = 0.8 - 0.4i

w_9 = i

w_10 = 3 - i

w_11 = 0.8 - 0.4i

w_12 = i

So the cycle is i → 3 - i → 0.8 - 0.4i → i → ... with period 3.

If this is the case, then starting from w_6, the sequence cycles every 3 steps. Let me verify this with exact calculations.

Let me compute w_6 = i. Then:

w_7 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i

w_8 = (1 - i) + 2i / (3 - i) = (1 - i) + 2i(3 + i)/10 = (1 - i) + (6i + 2i²)/10 = (1 - i) + (6i - 2)/10 = (1 - 2/10) + (-i + 6i/10) = 0.8 - 0.4i

w_9 = (1 - i) + 2i / (0.8 - 0.4i) = (1 - i) + 2i(0.8 + 0.4i)/ (0.64 + 0.16) = (1 - i) + (1.6i + 0.8i²)/0.8 = (1 - i) + (1.6i - 0.8)/0.8 = (1 - 0.8/0.8) + (-i + 1.6i/0.8) = (1 - 1) + (-i + 2i) = 0 + i = i

Yes, it cycles every 3 steps. So from w_6 onwards, the sequence is periodic with period 3: i, 3 - i, 0.8 - 0.4i, i, 3 - i, ...

Now, let me find the value of n where w_n enters this cycle. From earlier calculations:

w_0 = 1/137

w_1 = 1 + 273i

w_2 ≈ 1.0073 - 0.99997i

w_3 ≈ 0.00003 + 0.0073i

w_4 ≈ 275 - i

w_5 ≈ 1 - 0.9927i

w_6 = i

So w_6 is the first term in the cycle. Therefore, starting from n=6, the sequence is periodic with period 3.

Now, we need to find z_{2002}. Let me recall that z_n = w_n + i. So if I can find w_{2002}, then z_{2002} = w_{2002} + i.

First, let me determine how many steps it takes to reach the cycle. From n=0 to n=6, it takes 6 steps to reach the cycle. So for n ≥ 6, the sequence is periodic with period 3.

Therefore, for n ≥ 6, w_n = w_{6 + (n - 6)} = w_{6 + k}, where k = n - 6. Since the period is 3, w_{6 + k} = w_{6 + k mod 3}.

Therefore, to find w_{2002}, we need to compute k = 2002 - 6 = 1996. Then, k mod 3 = 1996 mod 3. Let me compute 1996 divided by 3: 3*665 = 1995, so 1996 = 3*665 + 1, so 1996 mod 3 = 1. Therefore, w_{2002} = w_{6 + 1} = w_7 = 3 - i.

Therefore, z_{2002} = w_{2002} + i = (3 - i) + i = 3.

Wait, this is a clean answer! Let me verify this. If w_{2002} = 3 - i, then z_{2002} = 3 - i + i = 3. So the answer is 3? Let me check with the cycle:

From n=6: w_6 = i → z_6 = i + i = 2i

Wait, no! Wait, z_n = w_n + i. So if w_6 = i, then z_6 = i + i = 2i. But earlier, when I computed w_6 = i, which was from w_6 = i, then z_6 = w_6 + i = i + i = 2i. But earlier, when I computed w_7 = 3 - i, then z_7 = 3 - i + i = 3. Similarly, w_8 = 0.8 - 0.4i, so z_8 = 0.8 - 0.4i + i = 0.8 + 0.6i. Then, w_9 = i, so z_9 = 2i, and so on.

But according to this, the cycle for z_n would be: z_6 = 2i, z_7 = 3, z_8 = 0.8 + 0.6i, z_9 = 2i, z_10 = 3, etc. So the period is 3 starting from n=6.

But according to our calculation, w_{2002} = w_7 = 3 - i, so z_{2002} = 3 - i + i = 3. Therefore, the answer is 3.

Let me verify this with the earlier steps. For example, z_7 = 3, z_8 = 0.8 + 0.6i, z_9 = 2i, z_10 = 3, etc. So if the period is 3 starting from n=6, then for n = 6 + 3k, z_n = 2i; n = 6 + 3k + 1, z_n = 3; n = 6 + 3k + 2, z_n = 0.8 + 0.6i.

Now, 2002 - 6 = 1996. 1996 divided by 3 is 665 with remainder 1, as 3*665 = 1995, 1996 = 1995 + 1. So 1996 = 3*665 + 1. Therefore, n = 6 + 1996 = 2002 corresponds to the first position in the cycle, which is z_7 = 3. Therefore, z_{2002} = 3.

This seems to be the case. Let me check with n=7: 7 - 6 = 1, which gives remainder 1, so z_7 = 3. Correct. Similarly, n=8: 8 - 6 = 2, remainder 2, z_8 = 0.8 + 0.6i. Correct.

Therefore, the answer is 3.

But let me confirm this with the earlier steps. Let me check if z_7 = 3. From earlier, w_7 = 3 - i, so z_7 = w_7 + i = 3 - i + i = 3. Yes.

Therefore, the final answer is 3.

**Final Answer**
\boxed{3}

To solve the problem, we are given a function:

$$
F(z) = \frac{z + i}{z - i}
$$

and an initial value:

$$
z_0 = \frac{1}{137} + i
$$

We are to compute $ z_{2002} $, where the sequence is defined by the recurrence:

$$
z_n = F(z_{n-1})
$$

---

### Step 1: Understanding the Function

The function $ F(z) $ is a Möbius transformation. These transformations are known to be invertible and can be composed, and their behavior under iteration can often be analyzed by finding fixed points or conjugating to simpler forms.

Let’s define a new variable:

$$
w_n = z_n - i
$$

Then, we can express the recurrence in terms of $ w_n $:

$$
z_n = w_n + i
$$

Substituting into the function:

$$
z_n = F(z_{n-1}) = \frac{z_{n-1} + i}{z_{n-1} - i}
$$

So:

$$
w_n + i = \frac{(w_{n-1} + i) + i}{(w_{n-1} + i) - i} = \frac{w_{n-1} + 2i}{w_{n-1}}
$$

Thus, the recurrence becomes:

$$
w_n = \frac{w_{n-1} + 2i}{w_{n-1}} = 1 + \frac{2i}{w_{n-1}}
$$

This is a recurrence relation for $ w_n $, which is simpler to analyze.

---

### Step 2: Compute Initial Terms

We are given:

$$
z_0 = \frac{1}{137} + i \Rightarrow w_0 = z_0 - i = \frac{1}{137}
$$

Now compute the first few terms of the $ w_n $ sequence:

- $ w_0 = \frac{1}{137} $
- $ w_1 = 1 + \frac{2i}{w_0} = 1 + 274i $
- $ w_2 = 1 + \frac{2i}{w_1} = 1 + \frac{2i}{1 + 274i} $

To simplify $ w_2 $, multiply numerator and denominator by the conjugate of the denominator:

$$
w_2 = 1 + \frac{2i(1 - 274i)}{(1 + 274i)(1 - 274i)} = 1 + \frac{2i - 548i^2}{1 + 274^2}
$$

Since $ i^2 = -1 $, this becomes:

$$
w_2 = 1 + \frac{2i + 548}{1 + 75076} = 1 + \frac{548 + 2i}{75077}
$$

So:

$$
w_2 = \frac{75077 + 548 + 2i}{75077} = \frac{75625 + 2i}{75077}
$$

This is a complex number close to 1, with a small imaginary part.

Continuing this process, we find that the sequence $ w_n $ eventually enters a **periodic cycle** starting at $ w_6 $:

$$
w_6 = i, \quad w_7 = 3 - i, \quad w_8 = 0.8 - 0.4i, \quad w_9 = i, \quad \text{and so on}
$$

So the sequence of $ w_n $ is periodic with period 3 starting from $ n = 6 $.

---

### Step 3: Determine $ w_{2002} $

We now compute the position of $ w_{2002} $ in the cycle.

Since the cycle starts at $ n = 6 $, we compute:

$$
2002 - 6 = 1996
$$

Now find the remainder when 1996 is divided by 3:

$$
1996 \mod 3 = 1
$$

So:

$$
w_{2002} = w_7 = 3 - i
$$

---

### Step 4: Convert Back to $ z_{2002} $

Recall that:

$$
z_n = w_n + i
$$

So:

$$
z_{2002} = w_{2002} + i = (3 - i) + i = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find z_{2002} where each term is defined by iterating the function F(z) = (z + i)/(z - i), starting from z_0 = 1/137 + i. Hmm, let me think about how to approach this.

First, since the function is being iterated, maybe there's a pattern or periodicity in the sequence z_n? If I can find the period, then I can figure out what z_{2002} is by finding the remainder when 2002 is divided by the period. But how do I find the period?

Alternatively, maybe I can find a closed-form expression for z_n. Let me try to analyze the function F(z). Let me write it down again:

F(z) = (z + i)/(z - i)

This looks like a Möbius transformation. Möbius transformations can sometimes be analyzed by finding their fixed points or by conjugating them to a simpler form, like a rotation or translation. Maybe I can find the fixed points of F(z)?

To find fixed points, set F(z) = z:

(z + i)/(z - i) = z

Multiply both sides by (z - i):

z + i = z(z - i)

Expand the right side:

z + i = z^2 - i z

Bring all terms to one side:

z^2 - i z - z - i = 0

Combine like terms:

z^2 - (1 + i) z - i = 0

Now, solve this quadratic equation for z. Let me compute the discriminant:

D = [-(1 + i)]^2 - 4 * 1 * (-i) = (1 + 2i + i^2) + 4i = (1 + 2i - 1) + 4i = 2i + 4i = 6i

Wait, discriminant is 6i? That's a complex number. Let me check my calculation again.

Original equation after moving terms:

z^2 - (1 + i) z - i = 0

Discriminant D = [-(1 + i)]^2 - 4 * 1 * (-i) = (1 + i)^2 + 4i

Compute (1 + i)^2: 1 + 2i + i^2 = 1 + 2i -1 = 2i. Then add 4i: 2i + 4i = 6i. Yes, that's correct. So D = 6i.

So solutions are:

z = [ (1 + i) ± sqrt(6i) ] / 2

Hmm, sqrt(6i)... Let me compute sqrt(i). Since i = e^{iπ/2}, so sqrt(i) = e^{iπ/4} = (1 + i)/√2. Therefore sqrt(6i) = sqrt(6) * e^{iπ/4} = sqrt(6)*(1 + i)/√2 = sqrt(3)*sqrt(2)*(1 + i)/√2 = sqrt(3)(1 + i). Wait, let me check:

sqrt(6i) = sqrt(6) * sqrt(i) = sqrt(6) * (1 + i)/√2 = sqrt(6)/sqrt(2) * (1 + i) = sqrt(3) * (1 + i). Yes, that's correct. So sqrt(6i) = sqrt(3)(1 + i). Therefore:

z = [ (1 + i) ± sqrt(3)(1 + i) ] / 2 = [ (1 + i)(1 ± sqrt(3)) ] / 2

Therefore, the fixed points are:

z = [ (1 + i)(1 + sqrt(3)) ] / 2 and z = [ (1 + i)(1 - sqrt(3)) ] / 2

Hmm, interesting. But I don't know if fixed points are directly helpful here. Maybe I need to consider the transformation properties. Alternatively, perhaps I can find an expression for z_n in terms of z_0 by iterating F.

Alternatively, maybe I can find an invariant or a transformation that simplifies the iteration. Let me think. Möbius transformations can be conjugated to translations or rotations if they have certain properties. Let me check if F(z) is a rotation or something else.

Alternatively, maybe I can use complex analysis. Let me consider writing z in terms of real and imaginary parts, but that might get messy. Alternatively, maybe consider using substitution. Let me try to find a substitution that linearizes the transformation.

Let me consider the function F(z) = (z + i)/(z - i). Let me try to find a substitution w(z) such that F(z) becomes a simpler function in terms of w. For example, if I can find a function w(z) such that w(F(z)) = w(z) + c or something like that, which would make the iteration easier.

Alternatively, since Möbius transformations can be represented by matrices, maybe I can find the matrix corresponding to F(z) and then find its powers. Let me recall that a Möbius transformation f(z) = (az + b)/(cz + d) corresponds to the matrix [[a, b], [c, d]], up to scalar multiples. So for F(z) = (z + i)/(z - i), the matrix would be [[1, i], [1, -i]]. Let me check:

Yes, because if you write f(z) = (az + b)/(cz + d), then a=1, b=i, c=1, d=-i. So the matrix is [[1, i], [1, -i]]. Now, if I can diagonalize this matrix or find its eigenvalues, then I can compute its powers more easily.

So let me find the eigenvalues of the matrix M = [[1, i], [1, -i]]. The eigenvalues λ satisfy det(M - λI) = 0:

|1 - λ    i        |
|1     -i - λ|  = 0

Compute determinant: (1 - λ)(-i - λ) - (i)(1) = 0

Expand:

(1 - λ)(-i - λ) - i = 0

Multiply out (1 - λ)(-i - λ):

= -i(1 - λ) - λ(1 - λ) = -i + iλ - λ + λ²

Then subtract i:

Total expression: -i + iλ - λ + λ² - i = λ² + iλ - λ - 2i = 0

So equation is λ² + (i - 1)λ - 2i = 0

Let me solve this quadratic equation for λ:

λ = [ -(i - 1) ± sqrt( (i - 1)^2 + 8i ) ] / 2

First compute (i - 1)^2:

(i - 1)^2 = (-1 + i)^2 = 1 - 2i + i² = 1 - 2i -1 = -2i

Then add 8i: -2i + 8i = 6i

So sqrt(6i) again. Wait, same as before. So sqrt(6i) is sqrt(3)(1 + i). Therefore:

λ = [ -(i - 1) ± sqrt(3)(1 + i) ] / 2

Simplify -(i - 1) = 1 - i. Therefore:

λ = [1 - i ± sqrt(3)(1 + i)] / 2

So the eigenvalues are:

λ₁ = [1 - i + sqrt(3)(1 + i)] / 2

and

λ₂ = [1 - i - sqrt(3)(1 + i)] / 2

Hmm, this seems complicated. Maybe there's a better way. Alternatively, perhaps I can find a function that conjugates F(z) to a simpler transformation. For example, if I can find a function w(z) such that w(F(z)) = w(z) + c, then iterating F would correspond to adding c each time. Let me try to find such a function.

Alternatively, since Möbius transformations can be conjugated to translations if they have a fixed point. Wait, if the transformation has two fixed points, then it can be conjugated to a translation. Let me recall that. If a Möbius transformation has two fixed points, then it's conjugate to a translation. So maybe I can conjugate F(z) to a translation, which would make the iteration easier.

So let me try that. Let me first find the fixed points again. Earlier, I found that the fixed points are z = [ (1 + i)(1 ± sqrt(3)) ] / 2. Let me denote them as α and β. Let me compute them numerically to get a sense.

First, compute sqrt(3) ≈ 1.732. So:

For α: [ (1 + i)(1 + sqrt(3)) ] / 2 ≈ [ (1 + i)(2.732) ] / 2 ≈ (2.732 + 2.732i)/2 ≈ 1.366 + 1.366i

For β: [ (1 + i)(1 - sqrt(3)) ] / 2 ≈ [ (1 + i)(-0.732) ] / 2 ≈ (-0.732 - 0.732i)/2 ≈ -0.366 - 0.366i

So the fixed points are approximately 1.366 + 1.366i and -0.366 - 0.366i. Let me denote them as α and β.

Now, if I can find a Möbius transformation that maps these fixed points to 0 and ∞, then the transformation would conjugate F(z) to a translation. Let me recall that if we have two fixed points α and β, then the transformation that maps α to 0 and β to ∞ is given by w(z) = (z - α)/(z - β). Then, the conjugated transformation would be w(F(z)) = w(z) + c for some constant c. Let me check.

Let me define w(z) = (z - α)/(z - β). Then, let me compute w(F(z)):

First, compute F(z) = (z + i)/(z - i). Then, w(F(z)) = [F(z) - α]/[F(z) - β] = [ (z + i)/(z - i) - α ] / [ (z + i)/(z - i) - β ]

Let me compute numerator and denominator:

Numerator: [ (z + i) - α(z - i) ] / (z - i) = [ z + i - α z + α i ] / (z - i) = [ (1 - α) z + i(1 + α) ] / (z - i)

Denominator: [ (z + i) - β(z - i) ] / (z - i) = [ z + i - β z + β i ] / (z - i) = [ (1 - β) z + i(1 + β) ] / (z - i)

Therefore, w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

But since w(z) = (z - α)/(z - β) = [ z - α ] / [ z - β ] = [ z - α ] / [ z - β ]

So if I want w(F(z)) = w(z) + c, then:

[ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] = [ z - α ] / [ z - β ] + c

This seems complicated. Maybe there's a better way. Alternatively, since we know that the transformation F(z) has fixed points α and β, then the conjugated transformation should be a translation. Let me recall that if we have a Möbius transformation with fixed points α and β, then it can be conjugated to a translation by the map w(z) = (z - α)/(z - β). Let me check this.

Let me denote the conjugated transformation as T(w) = w + c. Then, the composition would be:

w(F(z)) = T(w(z)) = w(z) + c

So, if I can find c such that this holds, then the iteration becomes w_{n+1} = w_n + c, which is easy to solve.

Let me try to compute this. Let me denote w(z) = (z - α)/(z - β). Then, we need:

w(F(z)) = w(z) + c

Let me compute w(F(z)):

As above, we have:

w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

But also, w(z) + c = [ (z - α)/(z - β) ] + c = [ (z - α) + c(z - β) ] / (z - β) = [ (1 + c) z - α - c β ] / (z - β)

Therefore, equating the two expressions:

[ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] = [ (1 + c) z - α - c β ] / (z - β)

Cross-multiplying:

[ (1 - α) z + i(1 + α) ] * (z - β) = [ (1 - β) z + i(1 + β) ] * [ (1 + c) z - α - c β ]

This equation must hold for all z (except the fixed points). Therefore, the coefficients of corresponding powers of z must be equal. This seems complicated, but maybe we can find c by choosing specific values or by using the fact that α and β are fixed points.

Alternatively, since α and β are fixed points of F(z), then F(α) = α and F(β) = β. Therefore, w(α) = 0 and w(β) = ∞. Then, if we consider the transformation T(w) = w + c, then applying T to w(F(z)) should give us w(z) + c. But since F(z) maps α to α, then w(F(α)) = w(α) = 0, and T(w(α)) = 0 + c = c. But also, w(F(α)) = w(α) + c = 0 + c = c. So that's consistent. Similarly for β, but since w(β) is ∞, adding c would still be ∞. So maybe this approach is not giving me the value of c directly.

Alternatively, maybe I can use the fact that the transformation F(z) is conjugate to a translation. Let me recall that if we have a Möbius transformation with two fixed points, then it's conjugate to a translation. The translation amount can be determined by the derivative at the fixed point or something else. Alternatively, since we have the matrix representation, maybe we can find the trace or something.

Wait, the matrix M = [[1, i], [1, -i]]. The trace of M is 1 + (-i) = 1 - i. The determinant is (1)(-i) - (i)(1) = -i - i = -2i. For Möbius transformations, the trace and determinant relate to the type of transformation. If the trace squared is greater than 4 times the determinant, it's hyperbolic, etc. But maybe I need to find the eigenvalues again. Wait, earlier I found that the eigenvalues are λ₁ and λ₂, which are [1 - i ± sqrt(3)(1 + i)] / 2. Let me compute their product. The product of eigenvalues is determinant of the matrix, which is -2i. So λ₁ * λ₂ = -2i. Also, the trace is λ₁ + λ₂ = 1 - i.

But how does this help me? Maybe if I can find the eigenvalues, then the matrix can be diagonalized, and then its powers can be computed. Let me try that. Suppose I can diagonalize M. Let me denote the eigenvalues as λ₁ and λ₂. Then, M^n = P D^n P^{-1}, where D is the diagonal matrix of eigenvalues. Then, the Möbius transformation corresponding to M^n would be the transformation that maps z to (a z + b)/(c z + d), where the matrix is M^n. However, I need to relate this to the iteration of F(z). Since each iteration corresponds to multiplying by M, then after n iterations, the transformation would be M^n. Therefore, if I can find M^n, then I can find the nth iterate of F(z).

But how do I compute M^n? Well, if I can diagonalize M, then it's straightforward. Let me try to find eigenvectors. Let me take the eigenvalues λ₁ and λ₂. Let me first compute λ₁ and λ₂ numerically to see if they are complex numbers. Let me compute λ₁:

λ₁ = [1 - i + sqrt(3)(1 + i)] / 2

Let me compute sqrt(3)(1 + i) ≈ 1.732*(1 + i) ≈ 1.732 + 1.732i

Then, adding 1 - i: 1 - i + 1.732 + 1.732i ≈ (1 + 1.732) + (-i + 1.732i) ≈ 2.732 + 0.732i

Divide by 2: ≈ 1.366 + 0.366i

Similarly, λ₂ = [1 - i - sqrt(3)(1 + i)] / 2 ≈ [1 - i - 1.732 - 1.732i]/2 ≈ (-0.732 - 2.732i)/2 ≈ -0.366 - 1.366i

So the eigenvalues are complex numbers. Therefore, the matrix M is diagonalizable with complex eigenvalues. Therefore, M^n can be expressed in terms of λ₁^n and λ₂^n. However, since the eigenvalues are complex, maybe they are conjugates? Let me check:

λ₁ ≈ 1.366 + 0.366i

λ₂ ≈ -0.366 - 1.366i

Wait, not sure if they are conjugates. Let me check if λ₂ is the conjugate of λ₁. The conjugate of λ₁ is 1.366 - 0.366i, which is not equal to λ₂. So they are not conjugates. Therefore, the matrix is not diagonalizable over real numbers, but over complex numbers. However, since we have two distinct eigenvalues, it is diagonalizable.

But maybe there's another approach. Let me recall that for Möbius transformations, if we have a transformation with two fixed points, then it can be conjugated to a translation. Let me try again with the fixed points α and β. Let me define w(z) = (z - α)/(z - β). Then, as before, w(F(z)) = w(z) + c. Let me try to compute c.

Let me compute w(F(z)) - w(z) = c. Let me compute this difference:

w(F(z)) - w(z) = [ (F(z) - α)/(F(z) - β) ] - [ (z - α)/(z - β) ] = c

Let me compute this expression. Let me denote F(z) = (z + i)/(z - i). Let me compute numerator and denominator:

First, compute F(z) - α = [(z + i)/(z - i) - α] = [ (z + i) - α(z - i) ] / (z - i) = [ z + i - α z + α i ] / (z - i) = [ (1 - α) z + i(1 + α) ] / (z - i)

Similarly, F(z) - β = [ (1 - β) z + i(1 + β) ] / (z - i)

Therefore, w(F(z)) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ]

Then, w(F(z)) - w(z) = [ (1 - α) z + i(1 + α) ] / [ (1 - β) z + i(1 + β) ] - [ (z - α)/(z - β) ] = c

This seems complicated, but maybe I can plug in z = α? Wait, but z = α is a fixed point, so F(α) = α, so w(F(α)) = w(α) = 0, and w(α) = 0, so 0 - 0 = c? That would imply c = 0, which can't be right. Wait, maybe not. Wait, if I plug z = α into the expression, then the first term is [ (1 - α) α + i(1 + α) ] / [ (1 - β) α + i(1 + β) ] and the second term is [ (α - α)/(α - β) ] = 0. Therefore, the difference is [ (1 - α) α + i(1 + α) ] / [ (1 - β) α + i(1 + β) ] = c. But since α is a fixed point, maybe we can use the fixed point equation?

Alternatively, maybe I can use the fact that F(z) = (z + i)/(z - i). Let me try to compute c by choosing a specific z. For example, take z = 0. Let me compute w(F(0)) - w(0) = c.

First, compute F(0) = (0 + i)/(0 - i) = i / (-i) = -1. Then, w(F(0)) = w(-1) = (-1 - α)/(-1 - β). Also, w(0) = (0 - α)/(0 - β) = (-α)/(-β) = α/β. Therefore, c = [ (-1 - α)/(-1 - β) ] - [ α/β ].

This seems messy, but maybe I can compute it using the expressions for α and β. Recall that α and β are fixed points of F(z), so they satisfy F(α) = α and F(β) = β. Let me recall that α and β satisfy the equation z = (z + i)/(z - i). Which led to the quadratic equation z² - (1 + i) z - i = 0. Therefore, α² - (1 + i) α - i = 0 and similarly for β.

Therefore, α² = (1 + i) α + i. Similarly for β².

Let me try to compute c using z = 0. Let me compute:

c = [ (-1 - α)/(-1 - β) ] - [ α/β ]

Let me compute numerator and denominator:

First term: (-1 - α)/(-1 - β) = (1 + α)/(1 + β)

Second term: α/β

Therefore, c = (1 + α)/(1 + β) - α/β

Let me combine these terms:

= [ (1 + α)β - α(1 + β) ] / [ β(1 + β) ]

Compute numerator:

(1 + α)β - α(1 + β) = β + α β - α - α β = β - α

Therefore, c = (β - α)/[ β(1 + β) ]

Hmm, interesting. So c = (β - α)/[ β(1 + β) ]

But I need to compute this. Let me recall that α and β are roots of the equation z² - (1 + i) z - i = 0. Therefore, from Vieta's formula:

α + β = 1 + i

α β = -i

Therefore, β - α can be found as follows. Let me compute (β - α)^2 = (α + β)^2 - 4 α β = (1 + i)^2 - 4*(-i) = (1 + 2i -1) + 4i = 2i + 4i = 6i. Therefore, β - α = sqrt(6i) or -sqrt(6i). Earlier, we saw that sqrt(6i) = sqrt(3)(1 + i). Therefore, β - α = sqrt(3)(1 + i) or -sqrt(3)(1 + i). Wait, but which one is it? Since α and β are the roots, depending on which is which. Let me check with the approximate values earlier. Earlier, α ≈ 1.366 + 1.366i and β ≈ -0.366 - 0.366i. Therefore, β - α ≈ (-0.366 - 0.366i) - (1.366 + 1.366i) ≈ -1.732 - 1.732i. Which is -sqrt(3)(1 + i). Therefore, β - α = -sqrt(3)(1 + i). Therefore, c = [ -sqrt(3)(1 + i) ] / [ β(1 + β) ]

Now, need to compute β(1 + β). Let me compute β(1 + β) = β + β². From the equation β² = (1 + i) β + i. Therefore, β + β² = β + (1 + i) β + i = (1 + i + 1) β + i = (2 + i) β + i. Wait, let me check:

Wait, β² = (1 + i) β + i, so β + β² = β + (1 + i) β + i = [1 + (1 + i)] β + i = (2 + i) β + i. Therefore, β(1 + β) = (2 + i) β + i.

But I need to compute this. Let me compute β. Recall that β is a root of z² - (1 + i) z - i = 0. Let me write β = [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, earlier we had sqrt( (1 + i)^2 + 4i ) = sqrt( -2i + 4i ) = sqrt(2i). Wait, no, earlier when solving for fixed points, we had discriminant D = 6i. Wait, maybe I need to compute β explicitly.

Alternatively, since α + β = 1 + i and α β = -i, then β = (1 + i) - α. Therefore, β(1 + β) = β + β². But β² = (1 + i) β + i, so β + β² = (1 + i) β + i + β = (2 + i) β + i. Therefore, β(1 + β) = (2 + i) β + i. Let me substitute β = [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, but this seems too complicated. Maybe there's another way.

Alternatively, since β satisfies β² = (1 + i) β + i, then β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this expression. Let me denote β(1 + β) = (2 + i) β + i. Let me compute this:

But I need to find β(1 + β). Let me use Vieta's formula. Since α + β = 1 + i and α β = -i. Let me compute β(1 + β) = β + β². From β² = (1 + i) β + i, so β + β² = β + (1 + i) β + i = (1 + i + 1) β + i = (2 + i) β + i. But I need to find this value. Let me think. Maybe I can express β in terms of α? Since β = (1 + i) - α. Therefore:

β(1 + β) = (2 + i) β + i = (2 + i)( (1 + i) - α ) + i. But this seems to bring me back to previous expressions. Maybe I need to compute numerically?

Alternatively, let me use the approximate value of β ≈ -0.366 - 0.366i. Let me compute β(1 + β):

First, compute 1 + β ≈ 1 + (-0.366 - 0.366i) ≈ 0.634 - 0.366i

Then, β(1 + β) ≈ (-0.366 - 0.366i)(0.634 - 0.366i)

Let me compute this:

Multiply the real parts: (-0.366)(0.634) ≈ -0.232

Multiply the cross terms: (-0.366)(-0.366i) + (-0.366i)(0.634) ≈ 0.134i - 0.232i ≈ -0.098i

Multiply the imaginary parts: (-0.366i)(-0.366i) = (0.134)i² = -0.134

So total ≈ -0.232 - 0.134 - 0.098i ≈ -0.366 - 0.098i

But this is approximate. However, I need an exact value. Let me try again with exact expressions.

Let me recall that β satisfies β² = (1 + i) β + i. Therefore, β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this:

(2 + i) β + i = (2 + i) β + i. Let me express β in terms of α. Since α + β = 1 + i, so β = 1 + i - α. Therefore:

(2 + i)(1 + i - α) + i = (2 + i)(1 + i) - (2 + i) α + i

Compute (2 + i)(1 + i) = 2(1) + 2i + i(1) + i(i) = 2 + 2i + i + i² = 2 + 3i -1 = 1 + 3i

Therefore, total expression is 1 + 3i - (2 + i) α + i = 1 + 3i + i - (2 + i) α = 1 + 4i - (2 + i) α

But α is another root, which satisfies α² = (1 + i) α + i. Also, α β = -i. Therefore, maybe I can express α in terms of β? Since α = (1 + i) - β. Therefore:

1 + 4i - (2 + i)(1 + i - β) = 1 + 4i - (2 + i)(1 + i) + (2 + i) β

Compute (2 + i)(1 + i) = 2(1) + 2i + i(1) + i(i) = 2 + 2i + i -1 = 1 + 3i, same as before. Therefore:

1 + 4i - (1 + 3i) + (2 + i) β = (1 + 4i - 1 - 3i) + (2 + i) β = i + (2 + i) β

But this seems to loop back. Maybe I need another approach. Let me recall that α β = -i. Therefore, β = -i / α. Therefore, substituting into β(1 + β):

β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i = (2 + i)(-i / α) + i = [ -i(2 + i) / α ] + i

Compute -i(2 + i) = -2i - i² = -2i + 1 = 1 - 2i. Therefore:

β(1 + β) = (1 - 2i)/α + i

But α is a root of z² - (1 + i) z - i = 0. Therefore, α² = (1 + i) α + i. Let me see if I can express 1/α in terms of α. From α² = (1 + i) α + i, divide both sides by α²:

1 = (1 + i)/α + i / α²

But this seems not helpful. Alternatively, from α² = (1 + i) α + i, rearrange:

α² - (1 + i) α = i

Divide both sides by α:

α - (1 + i) = i / α => α - (1 + i) = i / α => α - (1 + i) - i / α = 0. Not helpful.

Alternatively, maybe I can use the expression for β(1 + β) = (2 + i) β + i and then use β = -i / α:

β(1 + β) = (2 + i)(-i / α) + i = [ -i(2 + i) ] / α + i = [ (1 - 2i) ] / α + i

But from α² = (1 + i) α + i, we can solve for 1/α:

From α² = (1 + i) α + i => α² - (1 + i) α = i => α(α - (1 + i)) = i => α - (1 + i) = i / α => α = (1 + i) + i / α. Not helpful.

Alternatively, maybe I need to accept that this approach is too complicated and try another way. Let me recall that earlier, I had c = (β - α)/[ β(1 + β) ] and β - α = -sqrt(3)(1 + i). Therefore, c = -sqrt(3)(1 + i)/[ β(1 + β) ]

But if I can find β(1 + β), maybe I can relate it to something else. Let me think. From β² = (1 + i) β + i, then β(1 + β) = β + β² = β + (1 + i) β + i = (2 + i) β + i. Let me compute this expression:

(2 + i) β + i. Let me write β as [ (1 + i) - sqrt( (1 + i)^2 + 4i ) ] / 2. Wait, earlier we had the fixed points as [ (1 + i)(1 ± sqrt(3)) ] / 2. Wait, earlier when solving for fixed points, we had:

z = [ (1 + i)(1 ± sqrt(3)) ] / 2. Wait, but that was for the fixed points of F(z). Wait, but earlier I had that the fixed points satisfy z² - (1 + i) z - i = 0, which has solutions [ (1 + i) ± sqrt( (1 + i)^2 + 4i ) ] / 2. Let me compute (1 + i)^2 + 4i:

(1 + i)^2 = 1 + 2i + i² = 1 + 2i -1 = 2i. Then, adding 4i gives 6i. Therefore, sqrt(6i) = sqrt(3)(1 + i). Therefore, the fixed points are [ (1 + i) ± sqrt(3)(1 + i) ] / 2 = [ (1 + i)(1 ± sqrt(3)) ] / 2. Therefore, β is [ (1 + i)(1 - sqrt(3)) ] / 2. Therefore, β = (1 + i)(1 - sqrt(3))/2. Therefore, I can compute β(1 + β):

First, compute β = (1 + i)(1 - sqrt(3))/2. Let me compute 1 + β:

1 + β = 1 + (1 + i)(1 - sqrt(3))/2 = [ 2 + (1 + i)(1 - sqrt(3)) ] / 2

Let me compute (1 + i)(1 - sqrt(3)) = 1*(1 - sqrt(3)) + i*(1 - sqrt(3)) = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i)

Therefore, 1 + β = [ 2 + (1 - sqrt(3))(1 + i) ] / 2

Therefore, β(1 + β) = [ (1 + i)(1 - sqrt(3))/2 ] * [ 2 + (1 - sqrt(3))(1 + i) ] / 2

Let me compute this:

Let me denote A = (1 + i)(1 - sqrt(3))/2 and B = [ 2 + (1 - sqrt(3))(1 + i) ] / 2. Then β(1 + β) = A * B.

Let me compute A * B:

First, compute (1 - sqrt(3))(1 + i) = (1 - sqrt(3))(1 + i). Let me compute this:

= 1*(1 + i) - sqrt(3)*(1 + i) = (1 + i) - sqrt(3)(1 + i) = (1 - sqrt(3))(1 + i)

Wait, that's the same as before. Let me compute the value:

Let me compute 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me compute this term:

Let me compute (1 - sqrt(3))(1 + i) = (1)(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i). So 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me denote this as C = 2 + D, where D = (1 - sqrt(3))(1 + i).

Therefore, β(1 + β) = A * B = [ (1 + i)(1 - sqrt(3))/2 ] * [ (2 + D)/2 ] = [ (1 + i)(1 - sqrt(3)) * (2 + D) ] / 4

But D = (1 - sqrt(3))(1 + i), so 2 + D = 2 + (1 - sqrt(3))(1 + i). Let me compute this:

Let me compute 2 + (1 - sqrt(3))(1 + i). Let me expand (1 - sqrt(3))(1 + i):

= 1*(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i)

So 2 + (1 - sqrt(3))(1 + i) = 2 + (1 - sqrt(3))(1 + i). Let me compute this numerically? Maybe not. Let me think if there's a relation.

Alternatively, note that A = (1 + i)(1 - sqrt(3))/2 and B = [2 + (1 - sqrt(3))(1 + i)] / 2. Let me denote K = (1 - sqrt(3))(1 + i). Then, A = (1 + i)(1 - sqrt(3))/2 = K / 2. And B = [2 + K]/2. Therefore, β(1 + β) = (K / 2) * (2 + K)/2 = K(2 + K)/4.

So β(1 + β) = K(2 + K)/4, where K = (1 - sqrt(3))(1 + i). Let me compute K:

K = (1 - sqrt(3))(1 + i) = (1)(1) + 1*i - sqrt(3)*1 - sqrt(3)*i = 1 + i - sqrt(3) - sqrt(3)i = (1 - sqrt(3)) + i(1 - sqrt(3)) = (1 - sqrt(3))(1 + i). Wait, that's the same as before. Let me compute K numerically:

sqrt(3) ≈ 1.732, so 1 - sqrt(3) ≈ -0.732. Therefore, K ≈ (-0.732)(1 + i) ≈ -0.732 - 0.732i. Then, 2 + K ≈ 2 - 0.732 - 0.732i ≈ 1.268 - 0.732i. Then, K(2 + K) ≈ (-0.732 - 0.732i)(1.268 - 0.732i). Let me compute this:

First, multiply real parts: -0.732 * 1.268 ≈ -0.927

Cross terms: -0.732 * (-0.732i) + (-0.732i) * 1.268 ≈ 0.536i - 0.927i ≈ -0.391i

Imaginary parts: (-0.732i)(-0.732i) = (0.536)i² = -0.536

Total ≈ -0.927 - 0.536 - 0.391i ≈ -1.463 - 0.391i

Then, β(1 + β) ≈ (-1.463 - 0.391i)/4 ≈ -0.366 - 0.098i. Which matches the earlier approximate calculation.

But I need an exact value. Let me try to compute K(2 + K):

K = (1 - sqrt(3))(1 + i)

Let me compute K(2 + K):

= 2K + K²

First, compute 2K = 2(1 - sqrt(3))(1 + i)

Then, compute K² = [(1 - sqrt(3))(1 + i)]²

Let me compute K²:

= (1 - sqrt(3))² (1 + i)²

Compute (1 - sqrt(3))² = 1 - 2 sqrt(3) + 3 = 4 - 2 sqrt(3)

Compute (1 + i)² = 2i

Therefore, K² = (4 - 2 sqrt(3)) * 2i = 2i(4 - 2 sqrt(3)) = 8i - 4 sqrt(3) i

Therefore, K(2 + K) = 2K + K² = 2(1 - sqrt(3))(1 + i) + 8i - 4 sqrt(3) i

Let me compute 2(1 - sqrt(3))(1 + i):

= 2(1 + i - sqrt(3) - sqrt(3)i) = 2(1 - sqrt(3)) + 2i(1 - sqrt(3))

Therefore, K(2 + K) = 2(1 - sqrt(3)) + 2i(1 - sqrt(3)) + 8i - 4 sqrt(3) i

Combine like terms:

Real parts: 2(1 - sqrt(3))

Imaginary parts: 2(1 - sqrt(3))i + 8i - 4 sqrt(3) i

Let me compute the imaginary parts:

Factor out i:

[2(1 - sqrt(3)) + 8 - 4 sqrt(3)] i

Compute inside the brackets:

2(1 - sqrt(3)) + 8 - 4 sqrt(3) = 2 - 2 sqrt(3) + 8 - 4 sqrt(3) = 10 - 6 sqrt(3)

Therefore, K(2 + K) = 2(1 - sqrt(3)) + (10 - 6 sqrt(3))i

Therefore, β(1 + β) = K(2 + K)/4 = [2(1 - sqrt(3)) + (10 - 6 sqrt(3))i]/4 = [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2

Therefore, going back to c = (β - α)/[ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2

Wait, no. Earlier, c = (β - α)/[ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ β(1 + β) ] = [ -sqrt(3)(1 + i) ] / [ [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ] / 2 ] = [ -2 sqrt(3)(1 + i) ] / [ (1 - sqrt(3)) + (5 - 3 sqrt(3))i ]

This seems very complicated. Maybe I need to rationalize the denominator? Let me denote denominator as D = (1 - sqrt(3)) + (5 - 3 sqrt(3))i. Let me compute D:

Let me write D = a + bi, where a = 1 - sqrt(3), b = 5 - 3 sqrt(3). Then, c = [ -2 sqrt(3)(1 + i) ] / (a + bi). Multiply numerator and denominator by the conjugate of the denominator, which is a - bi:

c = [ -2 sqrt(3)(1 + i)(a - bi) ] / [ (a + bi)(a - bi) ] = [ -2 sqrt(3)(1 + i)(a - bi) ] / (a² + b²)

Compute denominator a² + b²:

a = 1 - sqrt(3), so a² = (1 - sqrt(3))² = 1 - 2 sqrt(3) + 3 = 4 - 2 sqrt(3)

b = 5 - 3 sqrt(3), so b² = (5)^2 - 2*5*3 sqrt(3) + (3 sqrt(3))² = 25 - 30 sqrt(3) + 27 = 52 - 30 sqrt(3)

Therefore, a² + b² = (4 - 2 sqrt(3)) + (52 - 30 sqrt(3)) = 56 - 32 sqrt(3)

Now, compute numerator: -2 sqrt(3)(1 + i)(a - bi). Let me compute (1 + i)(a - bi):

= 1*(a - bi) + i*(a - bi) = a - bi + ai - b i² = a - bi + ai + b = (a + b) + i(a - b)

Therefore, numerator becomes -2 sqrt(3) [ (a + b) + i(a - b) ]

Compute a + b = (1 - sqrt(3)) + (5 - 3 sqrt(3)) = 6 - 4 sqrt(3)

Compute a - b = (1 - sqrt(3)) - (5 - 3 sqrt(3)) = 1 - sqrt(3) - 5 + 3 sqrt(3) = -4 + 2 sqrt(3)

Therefore, numerator is -2 sqrt(3) [ (6 - 4 sqrt(3)) + i(-4 + 2 sqrt(3)) ] = -2 sqrt(3)(6 - 4 sqrt(3)) - 2 sqrt(3) i(-4 + 2 sqrt(3))

Let me compute each part:

First term: -2 sqrt(3)(6 - 4 sqrt(3)) = -12 sqrt(3) + 8 * 3 = -12 sqrt(3) + 24

Second term: -2 sqrt(3) * i * (-4 + 2 sqrt(3)) = 2 sqrt(3) i (4 - 2 sqrt(3)) = 8 sqrt(3) i - 4 * 3 i = 8 sqrt(3) i - 12 i

Therefore, numerator is (-12 sqrt(3) + 24) + (8 sqrt(3) - 12)i

Therefore, c = [ (-12 sqrt(3) + 24) + (8 sqrt(3) - 12)i ] / (56 - 32 sqrt(3))

This is getting extremely complicated. I must be missing a simpler approach. Let me think again.

Maybe instead of trying to find c, I can use the fact that if the transformation is conjugate to a translation, then the iteration becomes w_{n} = w_0 + n*c. Therefore, if I can find w_0 = w(z_0) and c, then w_{2002} = w_0 + 2002*c, and then z_{2002} = w^{-1}(w_{2002}).

But since I'm stuck on finding c, maybe I can try to compute w(z_0) and see if there's a pattern?

Let me recall that w(z) = (z - α)/(z - β). Given that z_0 = 1/137 + i, and α and β are the fixed points. Let me compute w(z_0):

w(z_0) = (z_0 - α)/(z_0 - β)

But without knowing α and β explicitly, this is difficult. However, maybe I can relate this to the earlier expressions. Let me recall that α and β are roots of z² - (1 + i) z - i = 0. Let me denote z_0 = 1/137 + i. Let me compute w(z_0):

Let me write w(z_0) = (z_0 - α)/(z_0 - β). Let me note that since α and β are roots of the quadratic equation, then (z - α)(z - β) = z² - (α + β) z + α β = z² - (1 + i) z - i. Therefore, for any z, (z - α)(z - β) = z² - (1 + i) z - i. Therefore, if I can express (z_0 - α)/(z_0 - β) in terms of z_0, maybe?

Let me denote w(z_0) = (z_0 - α)/(z_0 - β). Let me note that (z_0 - α)/(z_0 - β) = [ (z_0 - α) / (z_0 - β) ] = [ (z_0 - α) / (z_0 - β) ].

Alternatively, since (z_0 - α)(z_0 - β) = z_0² - (1 + i) z_0 - i, then (z_0 - α) = [ z_0² - (1 + i) z_0 - i ] / (z_0 - β). Therefore, w(z_0) = [ z_0² - (1 + i) z_0 - i ] / (z_0 - β)^2. Not sure if helpful.

Alternatively, maybe I can use the fact that if w(F(z)) = w(z) + c, then w(z_n) = w(z_0) + n*c. Therefore, if I can find w(z_0) and c, then I can find w(z_{2002}) = w(z_0) + 2002*c, and then invert the transformation to get z_{2002}.

But since I can't find c easily, maybe I can find c by using the fact that the transformation is conjugate to a translation. Let me recall that the transformation F(z) has a certain multiplier, which is the ratio of the eigenvalues or something. Wait, earlier I had the matrix M with eigenvalues λ₁ and λ₂. The trace of M is 1 - i, and the determinant is -2i. For Möbius transformations, the multiplier is the trace squared minus 4 times the determinant divided by something? Wait, no. The multiplier of a Möbius transformation is the derivative at a fixed point. Let me compute the derivative of F(z).

F(z) = (z + i)/(z - i). The derivative F’(z) = [ (1)(z - i) - (z + i)(1) ] / (z - i)^2 = [ z - i - z - i ] / (z - i)^2 = (-2i) / (z - i)^2. At a fixed point α, F’(α) = -2i / (α - i)^2. But since α is a fixed point, F(α) = α, so maybe I can relate this to the eigenvalues?

Alternatively, since the transformation is conjugate to a translation, the multiplier should be 1, since translations have derivative 1. Wait, but if it's conjugate to a translation, then the multiplier should be 1. However, the multiplier of F(z) is the derivative at a fixed point. Let me compute it.

Let me compute F’(z) = -2i / (z - i)^2. At the fixed point α, F’(α) = -2i / (α - i)^2. Let me compute α - i. Since α is a fixed point, α = (z + i)/(z - i) = z. Wait, no, α is a fixed point, so F(α) = α. Let me compute α - i. Let me recall that α satisfies α = (α + i)/(α - i). Let me solve for α - i:

Multiply both sides by α - i: α(α - i) = α + i

Expand: α² - i α = α + i

Rearrange: α² - i α - α - i = 0 => α² - (1 + i) α - i = 0, which is the original equation. Not helpful. Let me compute α - i:

From α = (α + i)/(α - i), rearrange:

α(α - i) = α + i => α² - i α = α + i => α² = (1 + i) α + i. Let me compute α - i:

Let me denote α - i = k. Then, α = k + i. Substitute into α² = (1 + i) α + i:

(k + i)^2 = (1 + i)(k + i) + i

Expand left side: k² + 2i k + i² = k² + 2i k - 1

Right side: (1 + i)k + (1 + i)i + i = (1 + i)k + i + i² + i = (1 + i)k + i - 1 + i = (1 + i)k + 2i - 1

Set equal:

k² + 2i k - 1 = (1 + i)k + 2i - 1

Subtract -1 from both sides:

k² + 2i k = (1 + i)k + 2i

Bring all terms to left:

k² + 2i k - (1 + i)k - 2i = 0 => k² + [2i - 1 - i]k - 2i = 0 => k² + (i - 1)k - 2i = 0

This is the same equation as before for λ! Wait, earlier when solving for eigenvalues, we had λ² + (i - 1)λ - 2i = 0. So this suggests that k = λ, the eigenvalues. Therefore, α - i = λ, so F’(α) = -2i / (α - i)^2 = -2i / λ². But since λ is an eigenvalue of the matrix M, and the trace of M is 1 - i, and the determinant is -2i, then for the eigenvalues λ₁ and λ₂, we have λ₁ + λ₂ = 1 - i and λ₁ λ₂ = -2i.

But if F’(α) = -2i / λ², and since the multiplier of the transformation is related to the eigenvalues, but I'm not sure. However, if the transformation is conjugate to a translation, then the multiplier should be 1, but here it's not necessarily. Wait, if it's conjugate to a translation, then the multiplier should be 1, but the multiplier here is F’(α) = -2i / λ². Let me compute this for λ₁.

Let me take λ₁ = [1 - i + sqrt(3)(1 + i)] / 2. Let me compute λ₁². This seems complicated, but maybe there's a relation. Since λ₁ and λ₂ are roots of λ² + (i - 1)λ - 2i = 0, then λ₁² = -(i - 1)λ₁ + 2i. Therefore, F’(α) = -2i / λ₁² = -2i / [ -(i - 1)λ₁ + 2i ] = -2i / [ (1 - i)λ₁ + 2i ].

This seems too complex. Maybe I need to accept that this approach is too difficult and try another way.

Let me go back to the original problem. We have z_0 = 1/137 + i, and we need to find z_{2002} under the iteration z_n = F(z_{n-1}).

Maybe I can compute the first few terms and see if there's a pattern or periodicity.

Let me compute z_0 = 1/137 + i.

Compute z_1 = F(z_0) = (z_0 + i)/(z_0 - i).

Let me compute numerator and denominator:

Numerator: z_0 + i = (1/137 + i) + i = 1/137 + 2i

Denominator: z_0 - i = (1/137 + i) - i = 1/137

Therefore, z_1 = (1/137 + 2i) / (1/137) = 1 + 2i * 137 = 1 + 274i

Wait, that's a big jump! So z_1 = 1 + 274i.

Now compute z_2 = F(z_1) = (z_1 + i)/(z_1 - i) = (1 + 274i + i)/(1 + 274i - i) = (1 + 275i)/(1 + 273i)

Let me rationalize this fraction. Multiply numerator and denominator by the conjugate of the denominator, which is 1 - 273i:

Numerator: (1 + 275i)(1 - 273i) = 1*1 + 1*(-273i) + 275i*1 + 275i*(-273i) = 1 - 273i + 275i - 275*273 i²

Simplify:

= 1 + 2i - 275*273*(-1) = 1 + 2i + 275*273

Compute 275*273: Let me compute 275*273. Note that 275 = 25*11, 273 = 3*7*13. Let me compute 275*273:

= (270 + 5)(270 + 3) = 270² + 270*3 + 5*270 + 5*3 = 72900 + 810 + 1350 + 15 = 72900 + 810 = 73710 + 1350 = 75060 + 15 = 75075. Wait, but this is for (270 + 5)(270 + 3). Wait, but 275*273 = (270 + 5)(270 + 3) = 270² + 270*3 + 5*270 + 5*3 = 72900 + 810 + 1350 + 15 = 72900 + 2175 = 75075. Yes.

So numerator = 1 + 2i + 75075 = 75076 + 2i

Denominator: (1 + 273i)(1 - 273i) = 1² - (273i)^2 = 1 - 273² * (-1) = 1 + 273²

Compute 273²: 273*273. Let me compute 270² + 2*270*3 + 3² = 72900 + 1620 + 9 = 74529. Therefore, denominator = 1 + 74529 = 74530

Therefore, z_2 = (75076 + 2i)/74530. Let me simplify this fraction. Let me divide numerator and denominator by 2:

Numerator: 75076 / 2 = 37538, 2i / 2 = i

Denominator: 74530 / 2 = 37265

So z_2 = (37538 + i)/37265. Let me see if this can be simplified further. Let me check if 37538 and 37265 have a common factor. Let me compute GCD(37538, 37265).

Compute 37538 - 37265 = 273. Now GCD(37265, 273).

Divide 37265 by 273: 273*136 = 273*(100 + 30 + 6) = 27300 + 8190 + 1638 = 27300 + 9828 = 37128. 37265 - 37128 = 137. Now GCD(273, 137).

273 ÷ 137 = 2 with remainder 273 - 2*137 = 273 - 274 = -1? Wait, 2*137 = 274, which is greater than 273. So 1*137 = 137, remainder 273 - 137 = 136. Now GCD(137, 136). Which is 1. Therefore, GCD is 1. So the fraction cannot be simplified further. Therefore, z_2 = (37538 + i)/37265.

This seems messy. Let me compute z_2 numerically to see if there's a pattern. Let me approximate:

Numerator: 37538 + i ≈ 37538 + 0i + i ≈ 37538 + i

Denominator: 37265

So z_2 ≈ (37538/37265) + (1/37265)i ≈ 1.0073 + 0.000027i. So approximately 1.0073 + 0.000027i. Very close to 1.

Wait, this is interesting. Let me check my calculation for z_2. Wait, z_1 = 1 + 274i. Then z_2 = (1 + 274i + i)/(1 + 274i - i) = (1 + 275i)/(1 + 273i). When I rationalized, I got numerator 75076 + 2i and denominator 74530. Let me check that again.

Wait, (1 + 275i)(1 - 273i) = 1*1 + 1*(-273i) + 275i*1 + 275i*(-273i) = 1 - 273i + 275i - 275*273*i². Since i² = -1, this becomes 1 + 2i + 275*273. Yes, and 275*273 = 75075, so total numerator is 1 + 2i + 75075 = 75076 + 2i. Denominator is 1 + 273² = 1 + 74529 = 74530. Correct. So z_2 ≈ 75076/74530 + (2/74530)i ≈ 1.0073 + 0.000027i. So very close to 1.

Now compute z_3 = F(z_2) = (z_2 + i)/(z_2 - i). Since z_2 is approximately 1 + 0.000027i, let me approximate z_2 ≈ 1 + εi, where ε is very small. Then:

z_3 ≈ (1 + εi + i)/(1 + εi - i) = (1 + i(ε + 1))/(1 - i(1 - ε)) ≈ [1 + i(1 + ε)] / [1 - i(1 - ε)]

If ε is very small, we can approximate this as [1 + i]/[1 - i] = (1 + i)^2 / (1 + 1) = (1 + 2i -1)/2 = 2i / 2 = i. So z_3 ≈ i. Let me check with exact values.

Wait, let me compute z_2 = (37538 + i)/37265. Let me compute z_2 + i = (37538 + i)/37265 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265

Therefore, z_3 = [37538 + 37266i] / [37538 - 37264i]

Let me rationalize this by multiplying numerator and denominator by the conjugate of the denominator, which is 37538 + 37264i:

Numerator: (37538 + 37266i)(37538 + 37264i)

Denominator: (37538)^2 + (37264)^2

This seems very complex, but let me note that the denominator is a large number, and the numerator might simplify. Let me see if there's a pattern.

Alternatively, since z_2 is very close to 1, let me consider that z_2 = 1 + ε, where ε is small. Then, z_3 = (1 + ε + i)/(1 + ε - i). Let me compute this:

= [ (1 + i) + ε ] / [ (1 - i) + ε ] ≈ [ (1 + i) + ε ] / (1 - i) * [1 - ε/(1 - i) + ... ] using the approximation 1/(1 + x) ≈ 1 - x for small x.

But this might not help. Alternatively, let me compute z_3 exactly. Let me note that z_2 = (37538 + i)/37265. Let me compute z_2 + i = (37538 + i)/37265 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265

Now, let me compute the numerator and denominator:

Numerator: 37538 + 37266i

Denominator: 37538 - 37264i

Let me note that 37538 = 37265 + 273. Let me write this as:

Numerator: (37265 + 273) + 37266i

Denominator: (37265 + 273) - 37264i

Let me factor out 37265 from numerator and denominator:

Numerator: 37265 + 273 + 37266i = 37265(1) + 273 + 37266i

Denominator: 37265 + 273 - 37264i = 37265(1) + 273 - 37264i

This doesn't seem helpful. Let me try to compute the ratio:

Let me denote A = 37538, B = 37266, C = 37538, D = 37264. Wait, no, numerator is A + Bi, denominator is A - Di, where A = 37538, B = 37266, D = 37264.

Let me compute the ratio (A + Bi)/(A - Di). Let me multiply numerator and denominator by A + Di:

Numerator: (A + Bi)(A + Di) = A² + ADi + ABi + BDi² = A² + (AD + AB)i - BD

Denominator: (A - Di)(A + Di) = A² + D²

So z_3 = [A² - BD + (AD + AB)i] / (A² + D²)

Let me compute A, B, D:

A = 37538

B = 37266

D = 37264

Compute A² - BD: Let me compute BD = 37266 * 37264. Note that 37266 * 37264 = (37265 + 1)(37265 - 1) = 37265² - 1

Similarly, A = 37538 = 37265 + 273

So A² = (37265 + 273)^2 = 37265² + 2*37265*273 + 273²

Therefore, A² - BD = 37265² + 2*37265*273 + 273² - (37265² - 1) = 2*37265*273 + 273² + 1

Let me compute this:

= 2*37265*273 + 273² + 1

= 273*(2*37265 + 273) + 1

Compute 2*37265 = 74530, so 74530 + 273 = 74803

Thus, A² - BD = 273*74803 + 1

This is a large number, but let me note that this is the real part of the numerator. The imaginary part is (AD + AB)i = A(D + B)i. Let me compute D + B = 37264 + 37266 = 74530. So the imaginary part is A * 74530 * i = 37538 * 74530 * i.

Therefore, z_3 = [273*74803 + 1 + 37538*74530 i] / (A² + D²)

This is extremely complex, but let me note that the denominator is A² + D² = (37538)^2 + (37264)^2. This is also a large number. However, given that z_2 was very close to 1, and z_3 is approximately i, maybe there's a pattern here. Let me check if z_3 is i.

If z_3 = i, then let me check if (z_2 + i)/(z_2 - i) = i. Let me solve for z_2:

(z_2 + i)/(z_2 - i) = i => z_2 + i = i(z_2 - i) => z_2 + i = i z_2 - i² => z_2 + i = i z_2 + 1 => z_2 - i z_2 = 1 - i => z_2(1 - i) = 1 - i => z_2 = 1.

But z_2 is approximately 1.0073, not exactly 1. So z_3 is not exactly i, but very close to it. Let me compute the exact value of z_3. Given that z_2 is very close to 1, let me assume that z_2 = 1 + ε, where ε is small. Then:

z_3 = (1 + ε + i)/(1 + ε - i) = [ (1 + i) + ε ] / [ (1 - i) + ε ] ≈ [ (1 + i) + ε ] / (1 - i) * [1 - ε/(1 - i) + ... ]

But this is getting too approximate. Alternatively, let me compute z_3 using the earlier expression:

z_3 = [A² - BD + (AD + AB)i] / (A² + D²). Let me compute A² - BD = 273*74803 + 1. Let me compute 273*74803. Let me note that 74803 = 74530 + 273. So 273*74803 = 273*74530 + 273^2.

But this is too tedious. Let me instead notice that if z_2 is very close to 1, then z_3 is very close to i, and then z_4 would be F(i) = (i + i)/(i - i) = 2i / 0, which is undefined. Wait, but z_2 is not exactly 1, so z_3 is not exactly i, but very close. Let me compute z_3 more accurately.

Given that z_2 ≈ 1 + ε, where ε = (37538 - 37265)/37265 + i*(1/37265) ≈ 273/37265 + i/37265 ≈ 0.0073 + 0.000027i. So ε ≈ 0.0073 + 0.000027i. Then, z_3 = (1 + ε + i)/(1 + ε - i) ≈ (1 + i + ε)/(1 - i + ε). Let me write this as [(1 + i) + ε]/[(1 - i) + ε]. Let me expand this using the approximation 1/(1 - i + ε) ≈ 1/(1 - i) * 1/(1 + ε/(1 - i)) ≈ 1/(1 - i) * [1 - ε/(1 - i) + ...].

So z_3 ≈ [(1 + i) + ε] * [1/(1 - i) - ε/(1 - i)^2 + ...]. Let me compute 1/(1 - i) = (1 + i)/2. So:

z_3 ≈ (1 + i) * (1 + i)/2 - ε * (1 + i)/2 * (1/(1 - i)^2) + ...

Wait, this is getting too complex. Let me compute the first term: (1 + i) * (1 + i)/2 = (1 + 2i -1)/2 = 2i/2 = i. The second term is -ε * (1 + i)/2 * [1/(1 - i)^2]. Let me compute 1/(1 - i)^2 = (1 + i)^2 / (1 + 1)^2 = (2i)/4 = i/2. Therefore, the second term is -ε * (1 + i)/2 * i/2 = -ε * (1 + i)i / 4 = -ε * (i - 1)/4. Since ε is small, this term is negligible. Therefore, z_3 ≈ i - negligible. So z_3 is approximately i.

If z_3 is approximately i, then z_4 = F(z_3) = (i + i)/(i - i) = 2i / 0, which is undefined. But since z_3 is not exactly i, let me compute it more precisely. Let me assume z_3 = i + δ, where δ is very small. Then, z_4 = F(z_3) = (z_3 + i)/(z_3 - i) = (i + δ + i)/(i + δ - i) = (2i + δ)/δ = 2i/δ + 1. If δ is very small, this would be a large number. But this suggests that the sequence might be approaching i, and then diverging. However, this seems inconsistent with the earlier steps.

Alternatively, maybe there's a cycle. Let me check if applying F multiple times brings us back to the original value. For example, if we start from z_0 = 1/137 + i, then z_1 = 1 + 274i, z_2 ≈ 1.0073 + 0.000027i, z_3 ≈ i, and then z_4 would be undefined? This seems problematic. Maybe there's a mistake in my calculations?

Wait, let me check z_3 again. If z_2 is very close to 1, then z_3 = (z_2 + i)/(z_2 - i) ≈ (1 + i)/(1 - i) = i. So z_3 ≈ i. Then z_4 = F(z_3) = (i + i)/(i - i) = 2i / 0, which is undefined. But since z_3 is not exactly i, let me compute it more precisely. Let me take z_2 = (37538 + i)/37265. Let me compute z_2 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265. Let me compute this as a complex number:

Real part: 37538 / 37265 ≈ 1.0073

Imaginary part: -37264 / 37265 ≈ -0.99997

So z_2 - i ≈ 1.0073 - 0.99997i ≈ 1.0073 - i. Wait, no, the imaginary part is -37264/37265 ≈ -1 + 1/37265 ≈ -1 + 0.000027. So z_2 - i ≈ 1.0073 - i + 0.000027i ≈ 1.0073 - i(1 - 0.000027). So approximately 1.0073 - i.

Then, z_2 + i ≈ 1.0073 + i + i = 1.0073 + 2i. Therefore, z_3 ≈ (1.0073 + 2i) / (1.0073 - i). Let me compute this:

Multiply numerator and denominator by the conjugate of the denominator, which is 1.0073 + i:

Numerator: (1.0073 + 2i)(1.0073 + i) ≈ (1.0073)^2 + 1.0073i + 2i*1.0073 + 2i^2

≈ 1.0146 + 1.0073i + 2.0146i - 2

≈ (1.0146 - 2) + (1.0073 + 2.0146)i

≈ -0.9854 + 3.0219i

Denominator: (1.0073)^2 + (1)^2 ≈ 1.0146 + 1 = 2.0146

So z_3 ≈ (-0.9854 + 3.0219i)/2.0146 ≈ -0.489 + 1.500i

Wait, this is different from my previous approximation. So z_3 ≈ -0.489 + 1.5i. Let me check if this is close to i? No, it's not. So my previous assumption was wrong. Let me compute this more accurately.

Wait, let me compute the exact value of z_3. Given that z_2 = (37538 + i)/37265, let me compute z_2 + i = (37538 + i + 37265i)/37265 = (37538 + 37266i)/37265

Similarly, z_2 - i = (37538 - 37264i)/37265

Let me compute the ratio:

Numerator: 37538 + 37266i

Denominator: 37538 - 37264i

Let me divide numerator and denominator by 37265:

Numerator: (37538/37265) + (37266/37265)i ≈ 1.0073 + 1.0000i

Denominator: (37538/37265) - (37264/37265)i ≈ 1.0073 - 0.99997i

So z_3 ≈ (1.0073 + i) / (1.0073 - i). Let me compute this:

Multiply numerator and denominator by (1.0073 + i):

Numerator: (1.0073 + i)^2 = 1.0073² + 2*1.0073*i + i² ≈ 1.0146 + 2.0146i - 1 = 0.0146 + 2.0146i

Denominator: (1.0073)^2 + 1 ≈ 1.0146 + 1 = 2.0146

So z_3 ≈ (0.0146 + 2.0146i)/2.0146 ≈ 0.00725 + i

So z_3 ≈ 0.00725 + i. This is very close to i, with a small real part. Let me call this z_3 ≈ i + ε, where ε = 0.00725. Then, z_4 = F(z_3) = (z_3 + i)/(z_3 - i) = (i + ε + i)/(i + ε - i) = (2i + ε)/ε = 2i/ε + 1. Since ε is small, this is a large number. Let me compute it:

z_4 ≈ 2i / 0.00725 + 1 ≈ 276i + 1. So z_4 ≈ 1 + 276i. Wait, this is similar to z_1 = 1 + 274i. Let me check if there's a pattern here.

Wait, z_1 = 1 + 274i, z_4 ≈ 1 + 276i. Let me compute z_5 = F(z_4) = (z_4 + i)/(z_4 - i) = (1 + 276i + i)/(1 + 276i - i) = (1 + 277i)/(1 + 275i). This is similar to z_2, which was (1 + 275i)/(1 + 273i). So it seems like there's a pattern where each time, the imaginary part increases by 2. Let me check:

z_1 = 1 + 274i

z_4 ≈ 1 + 276i

If this pattern continues, then z_7 would be 1 + 278i, and so on. If this is the case, then the sequence is periodic with period 3? Let me check.

Wait, from z_0 to z_1: +274i

z_1 to z_2: z_2 ≈ 1 + 0.00725i

z_2 to z_3: z_3 ≈ 0.00725 + i

z_3 to z_4: z_4 ≈ 1 + 276i

z_4 to z_5: similar to z_1 to z_2, leading to z_5 ≈ 1 + 0.00725i

Wait, this suggests that after three iterations, the sequence returns to a value close to z_2, and then repeats. If this is the case, then the period is 3. Let me check with exact calculations.

Let me assume that applying F three times brings us back to a value close to the original. Let me see:

If z_1 = 1 + 274i, then z_2 = (1 + 275i)/(1 + 273i), z_3 = (z_2 + i)/(z_2 - i) ≈ 0.00725 + i, and z_4 = F(z_3) ≈ 1 + 276i. Then, z_5 = F(z_4) = (1 + 276i + i)/(1 + 276i - i) = (1 + 277i)/(1 + 275i), which is similar to z_2, but with 277i instead of 275i. So it seems like each time, the imaginary part increases by 2. If this pattern continues, then after three iterations, the imaginary part increases by 2*3 = 6? Wait, from z_1 to z_4, the imaginary part increased by 2 (from 274i to 276i). So each iteration increases the imaginary part by 2? No, from z_1 to z_2, it decreased, then from z_2 to z_3, it increased, and from z_3 to z_4, it increased again. It's not clear.

Alternatively, let me notice that z_1 = 1 + 274i, z_4 ≈ 1 + 276i, which is 2 more. If this pattern continues, then z_7 would be 1 + 278i, and so on. If this is the case, then the sequence is not periodic, but the imaginary part increases by 2 every 3 steps. However, this seems unlikely.

Alternatively, let me consider that the transformation F(z) = (z + i)/(z - i) has a property that if we apply it three times, it might return to the original value or something. Let me try to compute F^3(z).

Let me recall that earlier, I tried to find the eigenvalues and the matrix, but it was too complex. Let me try to find a pattern with the first few terms.

We have:

z_0 = 1/137 + i

z_1 = 1 + 274i

z_2 = (1 + 275i)/(1 + 273i) ≈ 1.0073 + 0.000027i

z_3 ≈ 0.00725 + i

z_4 ≈ 1 + 276i

z_5 = (1 + 277i)/(1 + 275i) ≈ similar to z_2

z_6 ≈ 0.00725 + i

z_7 ≈ 1 + 278i

So it seems like every 3 steps, the imaginary part increases by 2. If this pattern continues, then the sequence is periodic with period 3, but with the imaginary part increasing. However, this seems contradictory. Alternatively, maybe there's a cycle where after three iterations, the value returns to a form similar to the original, but with a different constant.

Alternatively, let me consider that the transformation F(z) = (z + i)/(z - i) can be related to a rotation in the complex plane. Let me recall that if I write z = x + yi, then F(z) = (x + yi + i)/(x + yi - i) = (x + (y + 1)i)/(x + (y - 1)i). This resembles a Möbius transformation that might be a rotation or something else.

Alternatively, let me consider the transformation in terms of the imaginary part. Let me set w = z - i, then z = w + i. Substitute into F(z):

F(z) = (w + i + i)/(w + i - i) = (w + 2i)/w = 1 + 2i/w. So F(z) = 1 + 2i/w, where w = z - i.

Let me define w_n = z_n - i. Then, the recurrence becomes:

w_n = z_n - i = F(z_{n-1}) - i = [(z_{n-1} + i)/(z_{n-1} - i)] - i = [ (z_{n-1} + i) - i(z_{n-1} - i) ] / (z_{n-1} - i) = [ z_{n-1} + i - i z_{n-1} + i² ] / (z_{n-1} - i) = [ z_{n-1}(1 - i) + i - 1 ] / (z_{n-1} - i)

But since w_{n-1} = z_{n-1} - i, then z_{n-1} = w_{n-1} + i. Substitute this into the expression:

Numerator: (w_{n-1} + i)(1 - i) + i - 1

Let me compute this:

= w_{n-1}(1 - i) + i(1 - i) + i - 1

= w_{n-1}(1 - i) + i - i² + i - 1

= w_{n-1}(1 - i) + i + 1 + i - 1

= w_{n-1}(1 - i) + 2i

Therefore, w_n = [ w_{n-1}(1 - i) + 2i ] / (w_{n-1} + i - i) = [ w_{n-1}(1 - i) + 2i ] / w_{n-1} = (1 - i) + 2i / w_{n-1}

So the recurrence for w_n is:

w_n = (1 - i) + 2i / w_{n-1}

This is a recurrence relation for w_n. Let me write it as:

w_n = (1 - i) + 2i / w_{n-1}

This seems more manageable. Let me try to compute the first few terms.

Given z_0 = 1/137 + i, so w_0 = z_0 - i = 1/137.

Compute w_1 = (1 - i) + 2i / w_0 = (1 - i) + 2i / (1/137) = (1 - i) + 274i = 1 + 273i

Wait, earlier I had z_1 = 1 + 274i, so w_1 = z_1 - i = 1 + 274i - i = 1 + 273i. Correct.

Now compute w_2 = (1 - i) + 2i / w_1 = (1 - i) + 2i / (1 + 273i). Let me compute 2i / (1 + 273i). Multiply numerator and denominator by (1 - 273i):

2i(1 - 273i) / (1 + 273i)(1 - 273i) = [2i - 546i²] / (1 + 273²) = [2i + 546] / (1 + 74529) = (546 + 2i)/74530

Therefore, w_2 = (1 - i) + (546 + 2i)/74530 = [ (1 - i) * 74530 + 546 + 2i ] / 74530

Compute numerator:

1 * 74530 = 74530

- i * 74530 = -74530i

+ 546 + 2i

Total: 74530 + 546 + (-74530i + 2i) = 75076 - 74528i

Therefore, w_2 = (75076 - 74528i)/74530 = (75076/74530) - (74528/74530)i ≈ 1.0073 - 0.99997i, which matches our earlier approximation for z_2 - i.

Now compute w_3 = (1 - i) + 2i / w_2. Let me compute 2i / w_2 = 2i / [(75076 - 74528i)/74530] = 2i * 74530 / (75076 - 74528i) = 149060i / (75076 - 74528i). Let me rationalize this by multiplying numerator and denominator by (75076 + 74528i):

Numerator: 149060i * (75076 + 74528i) = 149060i * 75076 + 149060i * 74528i

Denominator: (75076)^2 + (74528)^2

This is very complex, but let me note that this is similar to previous calculations. However, let me notice that this is getting too cumbersome. Let me instead look for a pattern in the w_n sequence.

We have:

w_0 = 1/137

w_1 = 1 + 273i

w_2 = (75076 - 74528i)/74530 ≈ 1.0073 - 0.99997i

w_3 = (1 - i) + 2i / w_2

Let me assume that after some steps, the sequence enters a cycle. Let me try to compute w_3:

Let me denote w_2 = a - bi, where a = 75076/74530 ≈ 1.0073, b = 74528/74530 ≈ 0.99997. Then, 2i / w_2 = 2i / (a - bi) = 2i(a + bi)/ (a² + b²) = 2i(a + bi)/ (a² + b²) = 2i a / (a² + b²) - 2b / (a² + b²)

Therefore, w_3 = (1 - i) + 2i a / (a² + b²) - 2b / (a² + b²)

Let me compute a² + b². Since w_2 = (75076 - 74528i)/74530, then a² + b² = (75076² + 74528²)/74530². Let me compute this:

But this is equal to |w_2|² = |z_2 - i|². Since z_2 is close to 1, this is approximately |1 - i|² = 2. But let me compute it exactly:

From earlier, when we computed z_2 = (37538 + i)/37265, then w_2 = z_2 - i = (37538 + i)/37265 - i = (37538 + i - 37265i)/37265 = (37538 - 37264i)/37265. Therefore, |w_2|² = (37538² + 37264²)/37265². Let me compute 37538² + 37264². Note that 37538 = 37265 + 273, and 37264 = 37265 - 1. Let me compute:

(37265 + 273)^2 + (37265 - 1)^2 = 37265² + 2*37265*273 + 273² + 37265² - 2*37265*1 + 1 = 2*37265² + 2*37265*(273 - 1) + 273² + 1

This is complex, but let me note that this is equal to 2*37265² + 2*37265*272 + 273² + 1. However, this is not helpful. Let me instead note that this is equal to |w_2|² = |z_2 - i|². Since z_2 is close to 1, this is approximately |1 - i|² = 2. Let me assume that a² + b² ≈ 2. Then, w_3 ≈ (1 - i) + 2i a / 2 - 2b / 2 = (1 - i) + i a - b.

With a ≈ 1.0073 and b ≈ 0.99997, this becomes:

(1 - i) + i*1.0073 - 0.99997 ≈ (1 - 0.99997) + (-i + 1.0073i) ≈ 0.00003 + 0.0073i. So w_3 ≈ 0.00003 + 0.0073i. This is very small, close to 0.

Then, w_4 = (1 - i) + 2i / w_3. Since w_3 is very small, 2i / w_3 is very large. Let me approximate w_3 ≈ 0.0073i. Then, 2i / w_3 ≈ 2i / (0.0073i) = 2 / 0.0073 ≈ 274. So w_4 ≈ (1 - i) + 274 ≈ 275 - i.

Then, w_5 = (1 - i) + 2i / w_4 = (1 - i) + 2i / (275 - i). Let me compute 2i / (275 - i) = 2i(275 + i) / (275² + 1) ≈ 2i*275 / 275² = 2i / 275 ≈ 0.0073i. So w_5 ≈ (1 - i) + 0.0073i ≈ 1 - 0.9927i.

Then, w_6 = (1 - i) + 2i / w_5 ≈ (1 - i) + 2i / (1 - 0.9927i). Let me compute 2i / (1 - 0.9927i) ≈ 2i(1 + 0.9927i) / (1 + 0.9927²) ≈ 2i + 1.9854i² / 1.9854 ≈ 2i - 1.9854 / 1.9854 ≈ 2i - 1. So w_6 ≈ (1 - i) + 2i - 1 = (1 - 1) + (-i + 2i) = i.

Then, w_7 = (1 - i) + 2i / w_6 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i.

Then, w_8 = (1 - i) + 2i / w_7 = (1 - i) + 2i / (3 - i) = (1 - i) + 2i(3 + i)/ (9 + 1) = (1 - i) + (6i + 2i²)/10 = (1 - i) + (6i - 2)/10 = (1 - 2/10) + (-i + 6i/10) = 0.8 - 0.4i.

Then, w_9 = (1 - i) + 2i / w_8 = (1 - i) + 2i / (0.8 - 0.4i). Let me compute 2i / (0.8 - 0.4i) = 2i(0.8 + 0.4i) / (0.64 + 0.16) = (1.6i + 0.8i²) / 0.8 = (1.6i - 0.8)/0.8 = -1 + 2i. So w_9 ≈ (1 - i) + (-1 + 2i) = 0 + i.

Then, w_10 = (1 - i) + 2i / w_9 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i, which is the same as w_7. So it seems like from w_6 = i, w_7 = 3 - i, w_8 = 0.8 - 0.4i, w_9 = i, w_10 = 3 - i, and so on. This suggests a cycle of length 3 starting from w_6: i → 3 - i → 0.8 - 0.4i → i → ...

Wait, let me check:

w_6 = i

w_7 = 3 - i

w_8 = 0.8 - 0.4i

w_9 = i

w_10 = 3 - i

w_11 = 0.8 - 0.4i

w_12 = i

So the cycle is i → 3 - i → 0.8 - 0.4i → i → ... with period 3.

If this is the case, then starting from w_6, the sequence cycles every 3 steps. Let me verify this with exact calculations.

Let me compute w_6 = i. Then:

w_7 = (1 - i) + 2i / i = (1 - i) + 2 = 3 - i

w_8 = (1 - i) + 2i / (3 - i) = (1 - i) + 2i(3 + i)/10 = (1 - i) + (6i + 2i²)/10 = (1 - i) + (6i - 2)/10 = (1 - 2/10) + (-i + 6i/10) = 0.8 - 0.4i

w_9 = (1 - i) + 2i / (0.8 - 0.4i) = (1 - i) + 2i(0.8 + 0.4i)/ (0.64 + 0.16) = (1 - i) + (1.6i + 0.8i²)/0.8 = (1 - i) + (1.6i - 0.8)/0.8 = (1 - 0.8/0.8) + (-i + 1.6i/0.8) = (1 - 1) + (-i + 2i) = 0 + i = i

Yes, it cycles every 3 steps. So from w_6 onwards, the sequence is periodic with period 3: i, 3 - i, 0.8 - 0.4i, i, 3 - i, ...

Now, let me find the value of n where w_n enters this cycle. From earlier calculations:

w_0 = 1/137

w_1 = 1 + 273i

w_2 ≈ 1.0073 - 0.99997i

w_3 ≈ 0.00003 + 0.0073i

w_4 ≈ 275 - i

w_5 ≈ 1 - 0.9927i

w_6 = i

So w_6 is the first term in the cycle. Therefore, starting from n=6, the sequence is periodic with period 3.

Now, we need to find z_{2002}. Let me recall that z_n = w_n + i. So if I can find w_{2002}, then z_{2002} = w_{2002} + i.

First, let me determine how many steps it takes to reach the cycle. From n=0 to n=6, it takes 6 steps to reach the cycle. So for n ≥ 6, the sequence is periodic with period 3.

Therefore, for n ≥ 6, w_n = w_{6 + (n - 6)} = w_{6 + k}, where k = n - 6. Since the period is 3, w_{6 + k} = w_{6 + k mod 3}.

Therefore, to find w_{2002}, we need to compute k = 2002 - 6 = 1996. Then, k mod 3 = 1996 mod 3. Let me compute 1996 divided by 3: 3*665 = 1995, so 1996 = 3*665 + 1, so 1996 mod 3 = 1. Therefore, w_{2002} = w_{6 + 1} = w_7 = 3 - i.

Therefore, z_{2002} = w_{2002} + i = (3 - i) + i = 3.

Wait, this is a clean answer! Let me verify this. If w_{2002} = 3 - i, then z_{2002} = 3 - i + i = 3. So the answer is 3? Let me check with the cycle:

From n=6: w_6 = i → z_6 = i + i = 2i

Wait, no! Wait, z_n = w_n + i. So if w_6 = i, then z_6 = i + i = 2i. But earlier, when I computed w_6 = i, which was from w_6 = i, then z_6 = w_6 + i = i + i = 2i. But earlier, when I computed w_7 = 3 - i, then z_7 = 3 - i + i = 3. Similarly, w_8 = 0.8 - 0.4i, so z_8 = 0.8 - 0.4i + i = 0.8 + 0.6i. Then, w_9 = i, so z_9 = 2i, and so on.

But according to this, the cycle for z_n would be: z_6 = 2i, z_7 = 3, z_8 = 0.8 + 0.6i, z_9 = 2i, z_10 = 3, etc. So the period is 3 starting from n=6.

But according to our calculation, w_{2002} = w_7 = 3 - i, so z_{2002} = 3 - i + i = 3. Therefore, the answer is 3.

Let me verify this with the earlier steps. For example, z_7 = 3, z_8 = 0.8 + 0.6i, z_9 = 2i, z_10 = 3, etc. So if the period is 3 starting from n=6, then for n = 6 + 3k, z_n = 2i; n = 6 + 3k + 1, z_n = 3; n = 6 + 3k + 2, z_n = 0.8 + 0.6i.

Now, 2002 - 6 = 1996. 1996 divided by 3 is 665 with remainder 1, as 3*665 = 1995, 1996 = 1995 + 1. So 1996 = 3*665 + 1. Therefore, n = 6 + 1996 = 2002 corresponds to the first position in the cycle, which is z_7 = 3. Therefore, z_{2002} = 3.

This seems to be the case. Let me check with n=7: 7 - 6 = 1, which gives remainder 1, so z_7 = 3. Correct. Similarly, n=8: 8 - 6 = 2, remainder 2, z_8 = 0.8 + 0.6i. Correct.

Therefore, the answer is 3.

But let me confirm this with the earlier steps. Let me check if z_7 = 3. From earlier, w_7 = 3 - i, so z_7 = w_7 + i = 3 - i + i = 3. Yes.

Therefore, the final answer is 3.

**Final Answer**
\boxed{3}

To solve the problem, we are given a function:

$$
F(z) = \frac{z + i}{z - i}
$$

and an initial value:

$$
z_0 = \frac{1}{137} + i
$$

We are to compute $ z_{2002} $, where the sequence is defined by the recurrence:

$$
z_n = F(z_{n-1})
$$

---

### Step 1: Understanding the Function

The function $ F(z) $ is a Möbius transformation. These transformations are known to be invertible and can be composed, and their behavior under iteration can often be analyzed by finding fixed points or conjugating to simpler forms.

Let’s define a new variable:

$$
w_n = z_n - i
$$

Then, we can express the recurrence in terms of $ w_n $:

$$
z_n = w_n + i
$$

Substituting into the function:

$$
z_n = F(z_{n-1}) = \frac{z_{n-1} + i}{z_{n-1} - i}
$$

So:

$$
w_n + i = \frac{(w_{n-1} + i) + i}{(w_{n-1} + i) - i} = \frac{w_{n-1} + 2i}{w_{n-1}}
$$

Thus, the recurrence becomes:

$$
w_n = \frac{w_{n-1} + 2i}{w_{n-1}} = 1 + \frac{2i}{w_{n-1}}
$$

This is a recurrence relation for $ w_n $, which is simpler to analyze.

---

### Step 2: Compute Initial Terms

We are given:

$$
z_0 = \frac{1}{137} + i \Rightarrow w_0 = z_0 - i = \frac{1}{137}
$$

Now compute the first few terms of the $ w_n $ sequence:

- $ w_0 = \frac{1}{137} $
- $ w_1 = 1 + \frac{2i}{w_0} = 1 + 274i $
- $ w_2 = 1 + \frac{2i}{w_1} = 1 + \frac{2i}{1 + 274i} $

To simplify $ w_2 $, multiply numerator and denominator by the conjugate of the denominator:

$$
w_2 = 1 + \frac{2i(1 - 274i)}{(1 + 274i)(1 - 274i)} = 1 + \frac{2i - 548i^2}{1 + 274^2}
$$

Since $ i^2 = -1 $, this becomes:

$$
w_2 = 1 + \frac{2i + 548}{1 + 75076} = 1 + \frac{548 + 2i}{75077}
$$

So:

$$
w_2 = \frac{75077 + 548 + 2i}{75077} = \frac{75625 + 2i}{75077}
$$

This is a complex number close to 1, with a small imaginary part.

Continuing this process, we find that the sequence $ w_n $ eventually enters a **periodic cycle** starting at $ w_6 $:

$$
w_6 = i, \quad w_7 = 3 - i, \quad w_8 = 0.8 - 0.4i, \quad w_9 = i, \quad \text{and so on}
$$

So the sequence of $ w_n $ is periodic with period 3 starting from $ n = 6 $.

---

### Step 3: Determine $ w_{2002} $

We now compute the position of $ w_{2002} $ in the cycle.

Since the cycle starts at $ n = 6 $, we compute:

$$
2002 - 6 = 1996
$$

Now find the remainder when 1996 is divided by 3:

$$
1996 \mod 3 = 1
$$

So:

$$
w_{2002} = w_7 = 3 - i
$$

---

### Step 4: Convert Back to $ z_{2002} $

Recall that:

$$
z_n = w_n + i
$$

So:

$$
z_{2002} = w_{2002} + i = (3 - i) + i = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

Solution:
Iterating $F$ a few times, we get \[\begin{aligned} F(F(z)) &= \frac{\frac{z+i}{z-i}+i}{\frac{z+i}{z-i}-i} = \frac{(z+i)+i(z-i)}{(z+i)-i(z-i)}= \frac{z+i+zi+1}{z+i-zi-1}= \frac{(z+1)(i+1)}{(z-1)(1-i)}\\
&= \frac{(z+1)(i+1)^2}{(z-1) \cdot 2}= \frac{(z+1)(2i)}{(z-1) \cdot 2} = \frac{z+1}{z-1}i,\\
F(F(F(z))) &= \frac{\frac{z+1}{z-1}i+i}{\frac{z+1}{z-1}i-i} = \frac{\frac{z+1}{z-1}+1}{\frac{z+1}{z-1}-1} = \frac{(z+1)+(z-1)}{(z+1)-(z-1)}= z. \end{aligned}\]Thus, $z_{k+3} = z_k$ for all $k.$ Since $2002 \equiv 1 \pmod{3},$ we then have \[z_{2002} = z_1 = \frac{z_0+i}{z_0-i} = \frac{1/137 + 2i}{1/137} = \boxed{1+274i}.\]