Determine the multiplicity for the root z = 1 for

$\displaystyle f(z)=(1-cos2\pi z)^{30}$

How exactly?

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- Nov 22nd 2011, 10:43 AMSweatingBearDetermine multiplicity
Determine the multiplicity for the root z = 1 for

$\displaystyle f(z)=(1-cos2\pi z)^{30}$

How exactly? - Nov 22nd 2011, 11:11 AMgirdavRe: Determine multiplicity
Put $\displaystyle g(z)=1-\cos(2\pi z)$. Then $\displaystyle 1$ has multiplicity $\displaystyle 2$: write $\displaystyle g(z)=(z-1)^2h(z)$ where h is analytic and $\displaystyle h(1)\neq 0$. How can you conclude?

- Nov 22nd 2011, 11:25 AMSweatingBearRe: Determine multiplicity
- Nov 22nd 2011, 11:30 AMgirdavRe: Determine multiplicity
Well, since $\displaystyle f(z)=g(z)^{30}$, we can write $\displaystyle f(z)=(z-1)^{60}h(z)^{30}$, and since $\displaystyle h(1)^{30}$ we have the order. Hence the power 30 wasn't the main problem, you can try to do the exercise with $\displaystyle f(z)=(1-\cos (2\pi z))^p$ with $\displaystyle p$ a positive integer.