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| -rw-r--r-- | Fourier Series.page | 10 |
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 $f$ |