---
tag: E14W2V09HM
kind: concept
name: Residue calculus for Laplace inversion
msc: [44A10, "30E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Residue_theorem
---

## Summary

Residue calculus for Laplace inversion evaluates the [[Bromwich integral|5KNAXNHRY6]] with the [[residue theorem|032HKYS329]].
One closes the vertical line of integration with a large arc to the left.
When $F$ decays on such arcs, the [[inverse Laplace transform|07WBTX1MHA]] is the sum of the [[residues|0PHN07KVQT]] of $e^{s t} F(s)$ at the poles of $F$.
It turns inversion into a finite or infinite sum of explicit terms, one for each pole.

## Explanations

### L1 human

Each pole of the transform $F$ produces one piece of the original function of time.
A simple pole at $s = a$ produces a multiple of $e^{a t}$; a double pole produces terms like $t e^{a t}$.
For $F(s) = 1/(s^2 + 1)$, the poles $\pm i$ give $\frac{e^{i t}}{2 i} - \frac{e^{-i t}}{2 i} = \sin t$.
The method needs $F$ to be small on large arcs in the left half-plane; this fails for factors such as $e^{-s}$, which encode a time delay.