---
tag: HYRFMXAJVW
kind: concept
name: Laguerre series expansion of the Laplace transform
msc: [44A10, 33C45, 65R10]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre_polynomials
---

## Summary

The Laguerre series expansion writes the [[Laplace transform|0V49JEZW06]] $F(s)$ of a function, times $s + 1/2$, as a power series in $w = (s - 1/2)/(s + 1/2)$.
Here $w$ is a Möbius transformation of the transform variable $s$.
Its coefficients are the coefficients of the function in the orthonormal basis of Laguerre functions $e^{-t/2} L_n(t)$ of $L^2(0, \infty)$.
Here $L_n$ are the [[Laguerre polynomials|WAMRHWW82R]].
The expansion converts Laplace inversion into the computation of Taylor coefficients of an analytic function on the unit disk.
It is the basis of Weeks' method for the numerical inversion of the Laplace transform.

## Explanations

### L1 human

The Laplace transform of each Laguerre function is very simple: a power of $(s - 1/2)$ divided by a power of $(s + 1/2)$.
So if a function is a sum of Laguerre functions, its Laplace transform is a sum of these simple terms.
The change of variable $w = (s - 1/2)/(s + 1/2)$ makes this sum an ordinary power series in $w$.
For example, $f(t) = e^{-t/2}$ is the first Laguerre function, and its transform $1/(s + 1/2)$ is the first term of the series.
To recover $f$ from its transform, one reads off the power series coefficients in $w$.