We already states that functions with periodicity length \(L\) can be developed into a Fourier series: \begin{eqnarray*}f(x)&=&\sum_{n=-\infty}^\infty c_n e^{i\frac{2\pi}{L}nx}\quad\mbox{for period $L$}\\ c_n&=&\frac{1}{L}\int\limits_{-\frac{L}{2}}^{\frac{L}{2}} f(x)e^{-i\frac{2\pi}{L}nx} dx\end{eqnarray*}
Non periodic functions can be interpreted as functions with periodicity length \(L \to
\infty\).
We introduce a new variable \(k =\frac{2\pi}{L}n\), i.e. \(\Delta
k =\frac{2\pi}{L}\); so \(k\) becomes a contiuous variable for \(L \to \infty\). Replacing \(n\) by \(k\) in the above equations we get \begin{eqnarray*}f(x)&=&\frac{L}{2\pi}\sum_{n=-\infty}^\infty
c_k e^{i k x} \Delta k \quad\mbox{for period $L$}\\ c_k&=&\frac{1}{L}\int\limits_{-\frac{L}{2}}^{\frac{L}{2}} f(x)e^{-i
k x} dx\end{eqnarray*}
The decisive step now is to shift the variable \(L\) from the first equation into the second equation which as we will see in consequence translates the Kronecker-\(\delta\) into the \(\delta\) function: \begin{eqnarray*}f(x)&=&\frac{1}{2\pi}\sum_{n=-\infty}^\infty F(k) e^{i k(n) x} \Delta k \quad\mbox{for period $L$}\\ F(k)&=&\frac{L}{L}\int\limits_{-\frac{L}{2}}^{\frac{L}{2}} f(x)e^{-i k x} dx\end{eqnarray*}
Now the transition \(L \to \infty\) is possible, leading to \begin{eqnarray*}f(x)&=&\frac{1}{2\pi}\int\limits_{-\infty}^\infty F(k) e^{i k x} dk \quad\mbox{for non periodic functions}\\ F(k)&=&\int\limits_{-\infty}^\infty f(x)e^{-i k x} dx\end{eqnarray*}
Various versions of this equations exist, since the minus sign in the exponent can be shifted from the second into the first equation (, i.e. replacing \(k\) by \(-k\)) and the scaling factor \(\frac{1}{2\pi}\) can be shifted to the second equation or in a symmetrical definition the square root of this factor can be written in front of both integrals.
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© J. Carstensen (Math for MS)