Editing Dirac comb (section)

== Fourier series ==
{{see also|Dirichlet kernel}}
It is clear that <math>\operatorname{\text{Ш}}_{\ T}(t)</math> is periodic with period <math>T</math>. That is,
<math display="block">\operatorname{\text{Ш}}_{\ T}(t + T) = \operatorname{\text{Ш}}_{\ T}(t)</math>
for all ''t''. The complex Fourier series for such a periodic function is
<math display="block"> \operatorname{\text{Ш}}_{\ T}(t) = \sum_{n=-\infty}^{+\infty} c_n e^{i 2 \pi n \frac{t}{T}}, </math>
where (using [[Distribution_(mathematics)|distribution theory]]) the Fourier coefficients are
<math display="block">\begin{align}
c_n &= \frac{1}{T} \int_{t_0}^{t_0 + T} \operatorname{\text{Ш}}_{\ T}(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \quad ( -\infty < t_0 < +\infty ) \\
    &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \operatorname{\text{Ш}}_{\ T}(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \\
    &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \\
    &= \frac{1}{T} e^{-i 2 \pi n \frac{0}{T}} \\
    &= \frac{1}{T}.
\end{align}</math>

All Fourier coefficients are 1/''T'' resulting in
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} \!\!e^{i 2 \pi n \frac{t}{T}}.</math>

When the period is one unit, this simplifies to 
<math display="block">\operatorname{\text{Ш}}\ \!(x) = \sum_{n=-\infty}^{\infty} \!\!e^{i 2 \pi n x}.</math>
This is a [[divergent series]], when understood as a series of ordinary complex numbers, but becomes convergent in the sense of [[distribution (mathematics)|distributions]].

A "square root" of the Dirac comb is employed in some applications to physics, specifically:<ref>{{Cite book |last=Schleich |first=Wolfgang |title=Quantum optics in phase space |date=2001 |publisher=Wiley-VCH |isbn=978-3-527-29435-0 |edition=1st |pages=683–684}}</ref><math display="block">\delta_N^{(1 / 2)}(\xi) = \frac{1}{\sqrt{NT}} \sum_{\nu=0}^{N-1} e^{-i \frac{2\pi}{T}\xi \nu}, \quad 
\lim_{N \rightarrow \infty}\left|\delta_N^{(1 / 2)}(\xi)\right|^2= \sum_{k=-\infty}^{\infty} \delta(\xi - kT).</math>
However this is not a distribution in the ordinary sense.