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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): $\mathbf{N}$,$\mathbf{Z}$ |