quality:correction, quality:corruption, structural:reformatting
Integral representations

Often an integral representation of the Heaviside step function is useful: H (x) = lim ε → 0 + − 1 2 π i ∫ − ∞ ∞ 1 τ + i ε e − i x τ d τ = lim ε → 0 + 1 2 π i ∫ − ∞ ∞ 1 τ − i ε e i x τ d τ. {\displaystyle {\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau.\end{aligned}}}

where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.