From dfa84415288afae1f072229b074e6de926b82915 Mon Sep 17 00:00:00 2001 From: siveshs Date: Fri, 2 Jul 2010 20:18:29 +0000 Subject: section 2 editing --- Fourier Series.page | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) (limited to 'Fourier Series.page') diff --git a/Fourier Series.page b/Fourier Series.page index b16663f..eceb1ca 100644 --- a/Fourier Series.page +++ b/Fourier Series.page @@ -68,12 +68,13 @@ It is easy to show that any product of cosines and sines can be expressed as the As a final test to see if the Fourier series really could exist for any periodic function, we consider a periodic function with a sharp peak as shown below -![*Peak Function Image*](/peak_func.gif) +
![*Peak Function Image*](/peak_func.gif)
If it is possible to approximate the above function using a sum of sines and cosines, then it can be argued that *any* continuous periodic function can be expressed in a similar way. This is because any function could be expressed as a number of peaks at every position. It turns out that the above function can be approximated as the sum of two cosines, namely, $\cos^{2n}(x) + cos^{2n+1}(x)$ -
![alt text](/cos10x.gif) ![alt text](/cos11x.gif)
+
![$\cos^{2n}(x)$](/cos10x.gif) ![$cos^{2n+1}(x)$](/cos11x.gif)
+Summing these two ##What is the Fourier series actually? -- cgit v1.2.3