summaryrefslogtreecommitdiffstats
diff options
context:
space:
mode:
-rw-r--r--Problem Set 3.page26
1 files changed, 25 insertions, 1 deletions
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 \ No newline at end of file