section 3 editing

1 files changed, 9 insertions, 1 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
index b142534..d87cc22 100644
--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -167,7 +167,15 @@ $$
 (f, f_m) & = & \left( \Sigma a_n\,f_n , f_m \right)\\
 & = & \Sigma a_n\,\left(f_n , f_m \right)\\
 \end{array}
-$$
+$$  
+  
+Due to orthonormality of basis vectors, the inner product in the right-hand side of the above equation is $0$ for all terms except $n = m$.  
+Thus,  
+$$ (f, f_m) = a_m $$  
+Using the definition of the inner product,  
+$$ a_m = \int_0^{2\pi} \, f \, \frac{1}{sqrt{2\pi}} \, e^ {-inx} \, dx $$  
+  
+This is the common definition for the terms of the Fourier series.
 
 ##Proving that this function is does indeed completely represent $f$