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Problem 20

# Suppose that $$J: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$$ is defined by $$J(z)=1 / z, z \in \mathbb{C}_{\infty} .$$ Do our conventions imply $$J(0)=\infty$$ and $$J(\infty)=\infty$$ ? Does $$\chi(J(z), J(w))=\chi(z, w)$$ hold in $$\mathbb{C}_{\infty} ?$$

Expert verified
The conventions imply $$J(0)=\infty$$ and $$J(\infty)=0$$, not $$\infty$$. And $$\chi(J(z), J(w))$$ is not equal to $$\chi(z, w)$$ in the extended complex plane $$\mathbb{C}_{\infty}$$.
See the step by step solution

## Step 1: Evaluate function at 0 and $$\infty$$

To check if the conventions imply $$J(0)=\infty$$ and $$J(\infty)=\infty$$, substitute these values into the function $$J(z)=1 / z$$. For $$z=0$$, the function $$J$$ becomes $$J(0)=1/0$$, which is undefined in real arithmetic, but by convention in complex analysis, we consider this to be $$\infty$$. For $$z=\infty$$, the function $$J$$ becomes $$J(\infty)=1/\infty=0$$. Therefore, our conventions imply that $$J(0)=\infty$$ but $$J(\infty)=0$$ and not $$\infty$$.

## Step 2: Evaluate the cross-ratio

Compute the cross-ratio $$\chi(J(z), J(w))$$ and $$\chi(z, w)$$ to check if they are equal. The cross-ratio formula is given by $$\chi(z, w)=\frac{z-w}{1-z \cdot \overline{w}}$$. By applying the function $$J(z)=1/z$$ to $$z$$ and $$w$$, we have $$J(z)=1/z$$ and $$J(w)=1/w$$. Substituting these into the cross-ratio formula gives $$\mu(J(z), J(w))=\frac{1/z - 1/w}{1 - (1/z) \cdot (1/\overline{w})}$$. After simplifying, we get $$\mu(J(z), J(w))=\frac{\overline{w} - \overline{z}}{\overline{w} \cdot \overline{z} - 1}$$. This is, generally, not equal to $$\chi(z, w)$$. Thus $$\chi(J(z), J(w))$$ does not equal $$\chi(z, w)$$ in the extended complex plane $$\mathbb{C}_{\infty}$$.

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