Bromwich justification of the Heaviside calculus

Summary

The Bromwich justification gives the formal expressions $\varphi(p) H$ of the Heaviside operational calculus a precise meaning as contour integrals. It defines $\varphi(p) H$ by the Bromwich integral of $\varphi(s) e^{st} / s$ along a vertical line in the complex plane. With this definition, Heaviside's rules become theorems of complex analysis, and his expansion theorem becomes the residue theorem.