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 in ...

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