---
tag: 82YAXZMMQ1
kind: concept
name: Mikusiński field of operators
msc: [44A40]
sources:
  - url: https://en.wikipedia.org/wiki/Operational_calculus
---

## Summary

The Mikusiński field of operators is the [[field of fractions|0F2PTGX1A2]] of the ring of continuous functions on $[0, \infty)$ under addition and [[convolution on the half-line|FCKX8KFVGN]].
It exists because this ring has [[no zero divisors|E9QNNPQD2P]].
Its elements are called operators.
They include all such functions, all complex numbers, the Dirac delta as the unit $1$, and the differentiation operator $s$, the inverse of integration.

## Explanations

### L1 human

The rational numbers are built from the integers by allowing fractions $a/b$.
In the same way, Mikusiński built a field from continuous functions, with convolution as multiplication, by allowing fractions $f/g$.
Integration from $0$ is convolution with the constant function $1$, called $l$.
Its reciprocal $s = 1/l$ is not a function, but in this field it behaves like differentiation.