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| -rw-r--r-- | Problem Set 2.page | 2 |
1 files changed, 2 insertions, 0 deletions
diff --git a/Problem Set 2.page b/Problem Set 2.page index 523ce40..1cbcce1 100644 --- a/Problem Set 2.page +++ b/Problem Set 2.page @@ -62,6 +62,7 @@ $\int_0^{2\pi} \sin^4(x) dx = <1, f> = \sqrt{2\pi} \times < f_0, f >$ $= \sqrt{2\pi} \times a_0 = \frac{3 \pi}{4}$ + 3. Since $\sin x = \frac{e^{ix}-e^{-ix}}{2}$, @@ -86,6 +87,7 @@ Then, $\sum |a_n|^2 = {\sqrt{2\pi}/4}^2 + {- \sqrt{2\pi}/2}^2 + {\sqrt{2\pi}/4}^2 = \frac{3 \pi}{4}$ And, it was shown in Prob 2 that $\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$. + Therefore, $\int_0^{2\pi} \sin^4(x) dx = \sum |a_n|^2 = \frac{3 \pi}{4}$ |