Laplace's Integral

Author

Eric W. Weisstein
January 19, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/LaplacesIntegral.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Evaluation

v5.2

(expr1=(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π},GenerateConditions->False]//FullSimplify)//Timing

(expr2=(1/π)Integrate[(((x+(Sqrt[(x^2)-1])Cos[u]))^n),{u,0,π},GenerateConditions->False]//FullSimplify)//Timing

FullSimplify[expr1-expr2,n∈Integers&&Re[x]>0&&Im[x]>0]

Subtract@@@Table[{LegendreP[n,x],expr1}/.{n->Random[Integer,{-20,20}],x->Random[Complex,10(I+1){0,1},50]},{100}]//Chop//Union

Subtract@@@Table[{LegendreP[n,x],expr2}/.{n->Random[Integer,{0,20}],x->Random[Complex,10(I+1){-1,1},50]},{100}]//Chop//Union

V6.1

(expr1=(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π}]//FullSimplify)//Timing

V7

(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π}]//FullSimplify//Timing

(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π},Assumptions->x∈Reals]//FullSimplify

(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π},Assumptions->x>0]//FullSimplify

(1/π)Integrate[(1/(((x+(Sqrt[(x^2)-1])Cos[u]))^(n+1))),{u,0,π},Assumptions->x>0&&x≠1]//FullSimplify

FullSimplify[(((x-(Sqrt[-1+(x^2)])))^(-1-n)) Hypergeometric2F1[(1/2),1+n,1,-2 (-1+x (x+(Sqrt[-1+(x^2)])))]]

FunctionExpand[%]

Subtract@@@Table[{(((x-(Sqrt[-1+(x^2)])))^(-1-n)) Hypergeometric2F1[(1/2),1+n,1,-2 (-1+x (x+(Sqrt[-1+(x^2)])))],LegendreP[n,x]}/.{n->RandomInteger[{0,10}],x->RandomComplex[]},{1000}]//Chop//Union