From 3fbac26639134d806546c82626215c4bf4714385 Mon Sep 17 00:00:00 2001 From: siveshs Date: Fri, 2 Jul 2010 03:23:54 +0000 Subject: still testing --- Fourier Series.page | 11 +++++------ 1 file changed, 5 insertions(+), 6 deletions(-) diff --git a/Fourier Series.page b/Fourier Series.page index fdb70fc..a050d70 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -1,5 +1,3 @@ -Lecture on Fourier Series: - ##Why Fourier series possible? We first begin with a few basic identities on the size of sets. Show that the set of possible functions representing sets is not larger than the set of available functions? @@ -8,10 +6,11 @@ We first begin with a few basic identities on the size of sets. Show that the se 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} -x = 1 \\ -c = 2 -\end{array} +$$\begin{array}{ccl} +e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\ + & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\ + & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\ + & = & \cos y+i\sin y\end{array}$$ ##What is the Fourier series actually? -- cgit v1.2.3