Gaussian Functions

Gaussian Normal Distribution > s.a. fourier transform.
$ Normalized version:

F(x₀, ) = (2²)^–1/2 exp{–(x–x₀)²/2²} ;

In D dimensions, F(x₀, ) = (2²)^–D/2 exp{–(x–x₀)² / 2²}.

Gaussian Integrals
* Integrals of simple powers:

* Integrals of even powers:

* Remark: More can be obtained by taking derivatives wrt a; It may be possible to find a recursion formula.
@ References: Khrennikov qp/05 [integrals of analytic functionals on infinite-dimensional spaces].

Generalized Versions
* Complex exponent: By doing a contour integration over a pie-shaped wedge one can show that

_–infty^infty dx exp{–(a + ib)x²} = ^1/2 (a²+b²)^–1/4.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 5 jul 2008