##### The One-Dimensional Dirac Delta Function
c.m.dd.1d

- \(\delta(x-a) = 0\), \(x \neq a\).
- \(\int dx \delta (x - a) = 1\) if integration region includes \(x = a\), and vanishes otherwise.

Consequences: for any smooth differentiable function \(f(x)\),

- \(\int dx f(x) \delta (x - a) = f(a)\)
- \(\int dx f(x) \delta' (x-a) = -f'(a)\)
- \(\delta (f(x)) = \sum_i \frac{1}{\left| \frac{df}{dx} (x_i)\right|} \delta(x - x_i)\) where \(x_i\) are zeroes of \(f(x)\).

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42