Laplace transform of a Gaussian function

We evaluate the Laplace transformDlmfMathworldPlanetmath 11cf. Gaussian function,

{e-t2}=0e-ste-t2𝑑t=F(s). (1)

In fact,


By making the change of variable t+s2=u, we have (by the second equality in (1), the variable on operator’s argument is immaterial)


That is,


where erfc() is the complementary error functionDlmfDlmfPlanetmath. Its path of integration is subject to the restriction arg(u)θ, with |θ|π/4 as u along the path, with equality only if (u2) remains bounded to the left.

Title Laplace transform of a Gaussian function
Canonical name LaplaceTransformOfAGaussianFunction
Date of creation 2013-03-22 16:03:21
Last modified on 2013-03-22 16:03:21
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 5
Author perucho (2192)
Entry type Application
Classification msc 42-01