Gaussian Integral

Author

Eric W. Weisstein
September 21, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/GaussianIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GaussianIntegral.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Integrals

(∫_0^∞)(x^(2 s)) (e^(-a (x^2)))dx

v4.2.1

Integrate[x^(2s) Exp[-a x^2],{x,0,∞}]

Integrate[x^(2s) Exp[-a x^2],{x,0,∞},GenerateConditions->False]

v5.0.0

Integrate[x^(2s) Exp[-a x^2],{x,0,∞}]

Integrate[x^(2s) Exp[-a x^2],{x,0,∞},GenerateConditions->False]

v5.0.1

Integrate[x^(2s) Exp[-a x^2],{x,0,∞}]

Integrate[x^(2s) Exp[-a x^2],{x,0,∞},GenerateConditions->False]

Check

FullSimplify[FunctionExpand[(1/2) (a^(-(1/2)-s)) Gamma[(1/2)+s]-(2s-1)!!/2^(s+1)/a^s Sqrt[Pi/a]],a>0&&s∈Integers]

FunctionExpand[Gamma[s+1/2]-(s-1/2)!]

FullSimplify[(2s-1)!!/2^(s+1)/a^s Sqrt[Pi/a]==(s-1/2)!/2/a^(s+1/2),a>0&&s∈Integers]

Simplify[Table[{Integrate[x^(2s)Exp[-a x^2],{x,0,∞}],(2s-1)!!/2^(s+1)/a^s Sqrt[Pi/a],(s-1/2)!/2/a^(s+1/2)},{s,5}],a>0]

Subtract@@@Simplify[Table[{Integrate[x^n Exp[-a x^2],{x,0,∞},Assumptions->a>0],If[EvenQ[n],(n-1)!!/(2^(n/2+1)a^(n/2))Sqrt[Pi/a],((n-1)/2)!/2/a^((n+1)/2)]},
{n,5}],a>0]

(∫_0^∞)(e^(-a (x^2))) (x^n)dx

Table[Integrate[Exp[-a x^2]x^n,{x,0,∞},GenerateConditions->False],{n,10}]

Simplify[Table[If[EvenQ[n],(n-1)!!/2^(n/2+1)/a^(n/2) Sqrt[Pi/a],((n-1)/2)!/2/a^((n+1)/2)],{n,5}],a>0]

In Terms of (Subscript[H, n])(a)

FullSimplify[1/Sqrt[Pi]Integrate[Exp[-a x^2]x^n,{x,-∞,∞},GenerateConditions->False]]

Table[(((1+(((-1))^n)) (a^((1/2) (-1-n))) Gamma[((1+n)/2)])/(2 (Sqrt[π]))),{n,5}]

Table[HermiteH[n,a],{n,5}]

Brute Force

FullSimplify[Integrate[x^n Exp[-a x^2+b x],{x,-∞,∞},GenerateConditions->False]]

FullSimplify[%63,n∈Integers&&n>0]

FullSimplify[%,n∈Integers&&n>0]

eqn[n_]:=(I^n) (a^(-1-(n/2))) (E^((b^2)/(4 a))) ((Sqrt[a]) Cos[((n π)/2)] Gamma[((1+n)/2)] Hypergeometric1F1[-(n/2),(1/2),-((b^2)/(4 a))]-I b Gamma[1+(n/2)] Hypergeometric1F1[(1/2)-(n/2),(3/2),-((b^2)/(4 a))] Sin[((n π)/2)])

FullSimplify[Table[eqn[n],{n,3}]]

(∫_(-∞)^∞)(x^n) exp(b x-a (x^2))dx==(Sqrt[(π/a)]) exp(((b^2)/(4 a))) (∑_(k=0)^(n-1))((n! ((((1/(4 a))))^(n-k)) (((2 b))^(n-2 k)))/(k! (n-2 k)!))

V5

FullSimplify[Sqrt[Pi/a] Exp[b^2/4/a] Sum[n!/k!/(n-2k)!(1/4/a)^(n-k)(2b)^(n-2k),{k,0,n-1}],n∈Integers&&n>0]

FullSimplify[PowerExpand[%]]

eqn1[n_]:=(I^(-n)) (a^((1/2) (-1-n))) (E^((b^2)/(4 a))) (Sqrt[π]) HypergeometricU[-(n/2),(1/2),-((b^2)/(4 a))]

Simplify[PowerExpand[Table[eqn1[n],{n,3}]]]

V7

expr=Sqrt[Pi/a] Exp[b^2/4/a] Sum[n!/k!/(n-2k)!(1/4/a)^(n-k)(2b)^(n-2k),{k,0,n-1}]

expr=(2^(-n)) ((((1/a)))^((1/2)+n)) (b^n) (E^((b^2)/(4 a))) (Sqrt[π]) n! ((((2^(-1+n)) (((-((b^2)/a)))^((1/2)-(n/2))) HypergeometricU[(1/2)-(n/2),(3/2),-((b^2)/(4 a))])/(Gamma[1+n]))-((((((a/(b^2))))^n) HypergeometricPFQ[{1,(1/2)+(n/2),(n/2)},{1+n},((4 a)/(b^2))] Sin[n π])/(n π)));

FullSimplify[expr,n∈Integers&&n≥0&&a>0&&b∈Reals]

FullSimplify[expr,n∈Integers&&n≥0&&a>0&&b<0]

FullSimplify[expr,n∈Integers&&n≥0&&a>0&&b>0]

FullSimplify[(b^n)((Abs[b])^(-n)),n∈Integers&&n≥0&&b∈Reals]

FullSimplify[(I^(-n)) (a^(-(1/2)-(n/2))) (b^n) (E^((b^2)/(4 a))) (Sqrt[π]) ((Abs[b])^(-n)) ,n∈Integers&&n≥0&&b∈Reals]

FullSimplify[ (b^n) ((Abs[b])^(-n)) ,n∈Integers&&n≥0&&b∈Reals]

FullSimplify[ (b^n) ((Abs[b])^(-n)) HypergeometricU[-(n/2),(1/2),-((b^2)/(4 a))],n∈Integers&&n≥0&&b∈Reals]

(∫_(-∞)^∞)(x^n) (exp (b x-a (x^2)))dx==(Sqrt[(π/a)]) exp(((b^2)/(4 a))) (∑_(k=0)^(n-1))(2 k-1)!! (((2 a))^(k-n)) (b^(n-2 k)) (Binomial[n, 2 k])

V6

FullSimplify[Sqrt[Pi/a] Exp[b^2/4/a] Sum[Binomial[n,2k](2k-1)!!(2a)^(k-n)b^(n-2k),{k,0,n-1}]]

eqn2[n_]:=(Sqrt[(1/a)]) (E^((b^2)/(4 a))) (Sqrt[π]) (∑_(k=0)^(n-1))Binomial[n,2 k] (2 k-1)!! (((2 a))^(k-n)) (b^(n-2 k))

Simplify[PowerExpand[Table[eqn2[n],{n,3}]]]

V7

Sqrt[Pi/a] Exp[b^2/4/a] Sum[Binomial[n,2k](2k-1)!!(2a)^(k-n)b^(n-2k),{k,0,n-1}]

FullSimplify[%]