(1) Compute the cross-ratios (z, 1 + i, i, 2) and (z, 0, 1,∞), for z ∈ C∞.

To compute the cross-ratios for the given sets of complex numbers, we use the definition of the cross-ratio:

\[
(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}
\]

### 1. Cross-ratio \( (z, 1 + i, i, 2) \)

Given the points \( z \), \( 1 + i \), \( i \), and \( 2 \), the cross-ratio is:

\[
(z, 1 + i, i, 2) = \frac{(z - i)(1 + i - 2)}{(z - 2)(1 + i - i)}
\]

Now, simplifying the expressions inside the parentheses:

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

Substituting these into the formula:

\[
(z, 1 + i, i, 2) = \frac{(z - i)(-1 + i)}{(z - 2)(1)}
\]

Thus, the cross-ratio simplifies to:

\[
(z, 1 + i, i, 2) = \frac{(z - i)(-1 + i)}{z - 2}
\]

### 2. Cross-ratio \( (z, 0, 1, \infty) \)

The cross-ratio for \( (z, 0, 1, \infty) \) simplifies because one of the points is \( \infty \). When one of the points is \( \infty \), the cross-ratio reduces to the simpler form:

\[
(z, 0, 1, \infty) = \frac{z - 0}{z - 1} = \frac{z}{z - 1}
\]

Thus:

\[
(z, 0, 1, \infty) = \frac{z}{z - 1}
\]

### Summary:

1. \( (z, 1 + i, i, 2) = \frac{(z - i)(-1 + i)}{z - 2} \)
2. \( (z, 0, 1, \infty) = \frac{z}{z - 1} \)

Would you like more details on how cross-ratios work or help with simplifying further?

Here are 5 related questions for further practice:

1. How would you compute the cross-ratio \( (2, 1, 3, 0) \)?
2. Can the cross-ratio ever be a real number? If so, under what conditions?
3. How do Möbius transformations relate to cross-ratios?
4. What properties of cross-ratios remain invariant under transformations?
5. How is the cross-ratio used in projective geometry?

**Tip**: The cross-ratio is invariant under Möbius transformations, making it very useful in conformal mappings and projective geometry.

Compute the cross-ratios (z, 1 + i, i, 2) and (z, 0, 1,∞), for z ∈ C∞.

Learn how to compute the cross-ratios (z, 1+i, i, 2) and (z, 0, 1,∞) for complex numbers, using fundamental concepts in cross-ratio and complex geometry. This problem involves advanced mathematical ideas such as Möbius transformations and invariance properties.

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