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| -rw-r--r-- | Fourier Series.page | 3 |
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diff --git a/Fourier Series.page b/Fourier Series.page index a050d70..fb9ed7e 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -4,10 +4,9 @@ We first begin with a few basic identities on the size of sets. Show that the se ##Why Fourier series is plausible?</b> To show that Fourier series is plausible, let us consider some arbitrary trignometric functions and see if it is possible to express them as the sum of sines and cosines: - $1. \cos(2x) = 1 - 2 \sin^2(x)$ $$\begin{array}{ccl} -e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\ +\sin^2(x) & = & ?\\ & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\ & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\ & = & \cos y+i\sin y\end{array}$$ |