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Math Help - Analytic function

  1. #1
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    Analytic function

    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!
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  2. #2
    Moo
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    Hello,

    This may be useful
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  3. #3
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    Quote Originally Posted by bleys View Post
    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 CR 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.
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  4. #4
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    Quote Originally Posted by Moo
    Hello,

    This may be useful
    Ah thanks, that helps loads!

    Quote 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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