Laplace transform of a Gaussian function


We evaluate the Laplace transformDlmfMathworldPlanetmath 11cf. Gaussian function, wikipedia.org

{e-t2}=0e-ste-t2𝑑t=F(s). (1)

In fact,

{e-t2}=0e-(t2+2s2t+s24-s24)𝑑t=es240e-(t+s2)2𝑑t.

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)

{e-t2}=es24s2e-u2𝑑u.

That is,

{e-t2}=F(s)=π2es24erfc(s2),

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