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diff --git a/Problem Set 3.page b/Problem Set 3.page index 9504969..8f9876b 100644 --- a/Problem Set 3.page +++ b/Problem Set 3.page @@ -45,5 +45,29 @@ $$ f(z + \tau) = f(z) $$ 11. Compute the Fourier coefficients of the function $f(x) = \frac{1}{2} - \frac{x}{\pi}$ and use this to show that $$ \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$ +The proof of these facts -# Solutions +# Hints + +7. The Cauchy Riemann equations in polar coordinates are given by: + +$$ \frac{1}{r}\partial{f}{\partial \theta} = i \partial{f}{\partial r} $$ + +To derive this, you should use the chain rule, + +$$ \frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial \theta} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial \theta} $$ +$$ \frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial r} $$ + +together with the Cauchy-Riemann equations in rectangular coordinates. + +8. By applying Cauchy-Riemann equations in polar coordinates to a Fourier series +\[ f(r,\theta) = \sum_{n = -\infty}^{\infty} a_n(r) e^{in \theta} \] +you should obtain the following system of ordinary differential equations for the coefficients $a_n(r)$: +\[ \frac{d a_n}{dr} = \frac{na_n}{r} \] +Write this in the form +\[ \frac{d a_n}{a_n} = \frac{n dr}{r} \] +and integrate to get the solution. Then write +\[ z = re^{i\theta} \] +to derive the Laurent series. + +# Solutions
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