Analytic function

**bleys** · February 14th 2010, 05:05 AM

I've been trying to show that $1/1-z^4$ is analytic (whenever it's defined) using the Cauchy-Riemann equations. But I get some pretty ugly expressions that I can't simplify if I use $z = x + iy$ or polar form. Is there some way around it, or should I stick with that method and the expressions will simplify once I partially differentiate?
Thanks for any help!

**Moo** · February 14th 2010, 11:07 AM

Hello,

This may be useful

**Opalg** · February 14th 2010, 11:23 AM

Originally Posted by bleys

I've been trying to show that $1/1-z^4$ is analytic (whenever it's defined) using the Cauchy-Riemann equations. But I get some pretty ugly expressions that I can't simplify if I use $z = x + iy$ or polar form. Is there some way around it, or should I stick with that method and the expressions will simplify once I partially differentiate?
Thanks for any help!

Do you have to use the C–R equations? The easiest way to see that $f(z) = 1/(1-z^4)$ is analytic is to observe that by the ordinary rules for differentiation this function is differentiable, with $f'(z) = 4z^3/(1-z^4)^2$ .

**bleys** · February 14th 2010, 01:24 PM

Originally Posted by Moo

Hello,

This may be useful

Ah thanks, that helps loads!

Originally Posted by Opalg

Do you have to use the C–R equations? The easiest way to see that $f(z) = 1/(1-z^4)$ is analytic is to observe that by the ordinary rules for differentiation this function is differentiable, with $f'(z) = 4z^3/(1-z^4)^2$ .

Well, I just started complex analysis, so I want to get a hang of the C-R equations, even though it does say the differentiation rules are analogous to the real calculus ones. Thanks though!

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