In mathematics, Laguerre transform is an

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...

named after the mathematician

Edmond Laguerre Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigate ...

, which uses generalized

Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...

L_n^\alpha(x)

as kernels of the transform.McCully, Joseph. "The Laguerre transform." SIAM Review 2.3 (1960): 185-191. The Laguerre transform of a function

f(x)

is :

L\ = \tilde f_\alpha(n) = \int_^\infty e^ x^\alpha \ L_n^\alpha(x)\  f(x) \ dx

The inverse Laguerre transform is given by :

L^\ = f(x) = \sum_^\infty \binom^ \frac \tilde f_\alpha(n) L_n^\alpha(x)

Some Laguerre transform pairs

References

{{Reflist, 30em Integral transforms Mathematical physics