posted solutions of 2 and 3 in pset2

1 files changed, 15 insertions, 0 deletions

diff --git a/Problem Set 2.page b/Problem Set 2.page
index 5900d7f..5365ceb 100644
--- a/Problem Set 2.page
+++ b/Problem Set 2.page
@@ -38,6 +38,21 @@ $\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$
 
 # Solutions
 
+2. Since
+$sin(x) = \frac{e^{ix}-e^{-ix}}{2}$,
+$\int_0^{2\pi} \sin^4(x) dx = \frac{{e^{ix}-e^{-ix}}^4}{16}$
+$ = \frac{e^{i4x}+e^{-4ix}-4 e^{i2x} -4 e^{-i2x}+6}{16}$
+
+If we express any periodic function $f(x)$ as 
+$f(x) = \sum a_n f_n(x)$, where $f_n(x) = \frac{e^{inx}}{\sqrt{2\pi}}$, $f_0(x) = \frac{1}{\sqrt{2\pi}}$,
+
+The Fourier coefficients for the above functions are:
+$a_{-4} = a_{4} = \sqrt{2\pi} \times 1/16$, $a_{-2} = a_{2} = - \sqrt{2\pi} \times 4/16$, $a_0 = \sqrt{2\pi} \times 6/16$
+
+Since $a_m = < f_m, f >$,
+$\int_0^{2\pi} \sin^4(x) dx = <1, f> = \sqrt{2\pi} \times <f_0, f> = \frac{3 \pi}{4}$
+
+
 ## Cardinality
 
 Cardinality of the natural numbers (countable): $\mathbf{N}$,$\mathbf{Z}$