Pre-Quantum Electrodynamics

• \(\delta(x-a) = 0\), \(x \neq a\).
• \(\int dx \delta (x - a) = 1\) 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 \(f(x)\).