---
tag: 8050R3Z6HA
kind: concept
name: Operational calculus
msc: [44A45, 44A40, 44A10]
sources:
  - url: https://en.wikipedia.org/wiki/Operational_calculus
same_as:
  wikidata: Q1162633
  wikipedia: Operational calculus
---

## Summary

Operational calculus is a method that turns problems of analysis, above all linear differential and integral equations, into algebraic problems.
The differentiation operator is treated as an algebraic quantity, so an equation for the unknown function becomes an equation in a ring or field.
It is solved there by algebra, and the solution is translated back into a function.
The classical forms are the [[Heaviside operational calculus|QQ98C2AXZP]], its justification by the [[Laplace transform|0V49JEZW06]], and the [[Mikusiński operational calculus|TD5XK5AHNV]], which avoids integral transforms.

## Explanations

### L1 human

The idea is to treat "take the derivative" like multiplication by a number $s$.
Then a differential equation such as $y' - 2y = 0$ with $y(0) = 1$ becomes the algebra problem $sY - 1 - 2Y = 0$, so $Y = 1/(s - 2)$.
A table of translations says that $1/(s - 2)$ stands for the function $e^{2t}$, which is indeed the solution.
Engineers used this kind of symbolic shortcut long before mathematicians made it rigorous.