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| -rw-r--r-- | Fourier Series.page | 8 |
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: |