Geometric intepretation

We were given a complex number problem to work on: if $\mid s\mid =1$ , then show that $\Re(\frac{1}{s+1})=\frac{1}{2}$ I can prove this algebraically, but am having a little trouble with proving it geometrically. Can someone help me?

Easy Math Editor

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Comments

The transformation $s\mapsto \tfrac{1}{s+1}$ is a Mobius transformation, and Mobius transformations map circles/straight lines onto circles/straight lines. Thus the image of $|s|=1$ is a circle/straight line $L$ which passes through $\infty$ (the image of the point $-1$ on the original circle). Thus $L$ is a straight line. Since the point $2$ on the circle maps to the point $\tfrac12$ , $L$ passes through $\tfrac12$ .

Mobius transformations also preserve the angles between lines. The circle $|s|=1$ meets the real axis at right angles at $1$ . Hence $L$ meets the real axis (the image of the real axis) at right angles at $\tfrac12$ . Thus $L$ must be parallel to the imaginary axis, and we are done.

Mark Hennings - 7 years, 9 months ago

Thanks, appreciate it. :)

Edward Jiang - 7 years, 9 months ago

The set $A = \{s+1 \ : \ |s| = 1\}$ is a circle with center $1$ and radius $1$ . This circle passes through the origin. The set $A' = \{\tfrac{1}{s+1} \ : \ |s| = 1\}$ is the inversion of $A$ . Under inversion, circles passing through the origin map to lines that do not pass through the origin. Now, you just need to show that the line is given by $\mathfrak{R}(z) = \tfrac{1}{2}$ .

Jimmy Kariznov - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$