Chapter 2: Chapter 2

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 $\mathbb{C}_{\infty} ?$

Short Answer

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 $\mathbb{C}_{\infty}$.

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TABLE OF CONTENTS

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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Chapter 2: Chapter 2

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 \(\mathbb{C}_{\infty} ?\)

Short Answer

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Step 1: Evaluate function at 0 and \(\infty\)

Step 2: Evaluate the cross-ratio

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More chapters from the book ‘Complex Variables With Applications’

Chapter 1

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

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