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Diffstat (limited to 'Problem Set 3.page')
| -rw-r--r-- | Problem Set 3.page | 2 |
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diff --git a/Problem Set 3.page b/Problem Set 3.page index daae048..b1011b9 100644 --- a/Problem Set 3.page +++ b/Problem Set 3.page @@ -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: $$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$ |