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Re: Injective complex function proof
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Frederick Williams
Dec 23, 2009, 8:30:14 AM12/23/09
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John wrote:
>
> Prove that f(z) = z/(1-z)^2 is injective on unit circle.
>
> I know f can be written in the form
> z = r*e^(i*theta), r < 1.
> f(z) = r*e^(i*theta) / (1 - r*e^(i*theta))^2
> = (1-r*e^(i*theta))^(-1) (r^(-1) e^(-i*theta) - 1)^(-1)
>
> How should I continue from here ?
>
> Prove that f(z) = z/(1-z)^2 is injective on unit circle.
>
> I know f can be written in the form
> z = r*e^(i*theta), r < 1.
> f(z) = r*e^(i*theta) / (1 - r*e^(i*theta))^2
> = (1-r*e^(i*theta))^(-1) (r^(-1) e^(-i*theta) - 1)^(-1)
>
> How should I continue from here ?
If f is injective on the unit circle (call it U) then
for all a and b in U, if f(a) = f(b), then a = b.
Why not just apply this definition?
--
Pigeons were widely suspected of secret intercourse with the
enemy; counter-measures included the use of British birds of
prey to intercept suspicious pigeons in mid-air.
Christopher Andrew, 'Defence of the Realm', Allen Lane
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