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December 28th, 2014, 06:50 AM   #1
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Is finding laurent series expansion of f at z_0 using geometric series convenient?

Hello there,

I was taught in my complex analysis course that I can find the laurent series expansion for $f:\Omega \to \mathbb{C}$ by using the formula for geometric series:

f(z) = \frac{1}{1-g(z)} = \sum_{n=0}^\infty g(z)^n

This is useful for series expanded at $z=0$. However, I am interested in finding convenient methods for making laurent expansions at at other points $z_0$. I would like to make series of the form:

$$f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n,$$

where $z_0$ may be different for $0$.

Is it convenient to use the geometric series construction to do this? If so, how? And if not, which are the convenient methods to do so? I know that the coefficients are given by $a_n = 1/(2\pi i) \oint_C f(z)/(z-z_o)^{n+1} \mathrm{d}z$, but it is tedious to calculate all the coeffients manually.

Thank you for your time.

Kind regards,

Last edited by king.oslo; December 28th, 2014 at 07:44 AM.
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