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| -rw-r--r-- | Problem Set 3.page | 8 |
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diff --git a/Problem Set 3.page b/Problem Set 3.page index 8f9876b..e5bac92 100644 --- a/Problem Set 3.page +++ b/Problem Set 3.page @@ -61,13 +61,13 @@ $$ \frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}\frac{\partial x together with the Cauchy-Riemann equations in rectangular coordinates. 8. By applying Cauchy-Riemann equations in polar coordinates to a Fourier series -\[ f(r,\theta) = \sum_{n = -\infty}^{\infty} a_n(r) e^{in \theta} \] +$$ f(r,\theta) = \sum_{n = -\infty}^{\infty} a_n(r) e^{in \theta} $$ you should obtain the following system of ordinary differential equations for the coefficients $a_n(r)$: -\[ \frac{d a_n}{dr} = \frac{na_n}{r} \] +$$ \frac{d a_n}{dr} = \frac{na_n}{r} $$ Write this in the form -\[ \frac{d a_n}{a_n} = \frac{n dr}{r} \] +$$ \frac{d a_n}{a_n} = \frac{n dr}{r} $$ and integrate to get the solution. Then write -\[ z = re^{i\theta} \] +$$ z = re^{i\theta} $$ to derive the Laurent series. # Solutions
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