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authorluccul <luccul@gmail.com>2010-07-06 05:00:48 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-07-06 05:00:48 +0000
commit62b3ad25eee24cf5fe91be19c4e99eb38a70c08b (patch)
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parent1eca287d3f83cf9133e42c644f9d809792f32cec (diff)
downloadafterklein-wiki-62b3ad25eee24cf5fe91be19c4e99eb38a70c08b.tar.gz
afterklein-wiki-62b3ad25eee24cf5fe91be19c4e99eb38a70c08b.zip
Aha. $1$.
-rw-r--r--Problem Set 3.page2
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@@ -32,7 +32,7 @@ Use Fourier series to solve the wave equation in the case of a vibrating ring.
$$ f(r,\theta) = \sum_n a_n(r)e^{in\theta} $$
as a Fourier series whose coefficients depend on $r$. Use the Cauchy-Riemann equations in polar coordinates to derive the Laurent series expansion of $f$ without using the map $z \mapsto e^{iz}$.
-9. Let $f(z)$ be a holomorphic function defined on a region that contains the disk of radius $1$. Derive the following variant of the Cauchy integral formula:
+9. Let $f(z)$ be a holomorphic function defined on a region that contains the disk of radius 1. Derive the following variant of the Cauchy integral formula:
$$ f(z) = \frac{1}{2\pi} \int_0^{2\pi} \frac{f(e^{i\theta})}{1 - e^{-i\theta}z} d\theta $$
Hint: Expand the right hand side using the formula for a geometric series:
$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$