section 3 editing

1 files changed, 6 insertions, 2 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
index ce33109..ae34c0f 100644
--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -121,9 +121,13 @@ In order to prove orthonormality of the basis vectors:
   
 $$
 \begin{array}{ccl}
-(f_n,f_m) = \int_0^{2\pi} \, \frac{1}{\sqrt{2\pi}} \, e^{inx} \, \bar {\frac{1}{\sqrt{2\pi}} \, e^{inx}} \, dx\\
+(f_n,f_m) = \int_0^{2\pi} \, \frac{1}{\sqrt{2\pi}} \, e^{inx} \, \longbar {\frac{1}{\sqrt{2\pi}} \, e^{inx}} \, dx\\
+& = & \frac{1}{2\pi} \, \int_0^{2\pi} \, e^{i(n-m)x} \, dx \\
+Here, n = m \Rightarrow (f_n,f_m) & = & 1\\
+n \neq m \Rightarrow (f_n,f_m) & = & 0\\
 \end{array}
-$$ 
+$$  
+  
   
 ##Determining Coefficients of the Basis vectors
 In any vector space, the inner product of a vector and its basis vector gives the coefficient. For example, consider a 2-dimensional vector as shown below: