1. ## Analytic function

I've been trying to show that $\displaystyle 1/1-z^4$ 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 $\displaystyle z = x + iy$ 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!

2. Hello,

This may be useful

3. Originally Posted by bleys
I've been trying to show that $\displaystyle 1/1-z^4$ 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 $\displaystyle z = x + iy$ 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 $\displaystyle f(z) = 1/(1-z^4)$ is analytic is to observe that by the ordinary rules for differentiation this function is differentiable, with $\displaystyle f'(z) = 4z^3/(1-z^4)^2$.

4. Originally Posted by Moo
Hello,

This may be useful

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