---
tag: GD8B883592
kind: concept
name: Bromwich justification of the Heaviside calculus
msc: [44A45, 44A10]
sources:
  - url: https://en.wikipedia.org/wiki/Operational_calculus
---

## Summary

The Bromwich justification gives the formal expressions $\varphi(p) H$ of the [[Heaviside operational calculus|QQ98C2AXZP]] a precise meaning as contour integrals.
It defines $\varphi(p) H$ by the [[Bromwich integral|5KNAXNHRY6]] 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.

## Explanations

### L1 human

Heaviside's symbolic tricks worked but nobody could say why.
The answer was to read each symbolic expression as an integral in the complex plane.
Evaluating that integral by residues reproduces exactly the answers that Heaviside's rules give.
For example, $p^{-1} H$ becomes an integral whose only residue gives the function $t$, the integral of the unit step.