Glasser's Master Theorem

Author

Eric W. Weisstein
April 14, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Cauchy's Example

Integrate[Exp[-x^2],{x,-∞,∞}]

Integrate[Exp[-u^2]/. u->x-1/x,{x,-∞,∞}]

Glasser's Example

Correct value

Should be 2π independent of a and b

v4.2

Wrong in v4.2

$Version

int=Integrate[((((2x+a+b))^2)/((((x+a))^2)(((x+b))^2)+(((2x+a+b))^2))),{x,-∞,∞},PrincipalValue->True];

int2=FullSimplify[PowerExpand[int]]

ran=Table[Random[],{10},{2}]

{#1,#2,int /.{a->#1,b->#2}}&@@@ran

{#1,#2,int2 /.{a->#1,b->#2}}&@@@ran

NIntegrate[Evaluate[((((2x+a+b))^2)/((((x+a))^2)(((x+b))^2)+(((2x+a+b))^2)))/.Thread[{a,b}->#]],{x,-∞,∞},MaxRecursion->30,Method->GaussKronrod]&/@ran

2.Pi

int3=Integrate[((((2x+a+b))^2)/((((x+a))^2)(((x+b))^2)+(((2x+a+b))^2))),{x,-∞,∞},Assumptions->{a>0,b>0}];

FullSimplify[PowerExpand[int3]]

int4=Integrate[((((2x+a+b))^2)/((((x+a))^2)(((x+b))^2)+(((2x+a+b))^2))),{x,-∞,∞},PrincipalValue->True,Assumptions->{a>0,b>0}];

FullSimplify[PowerExpand[int4]]

v5.0

Correct, but not fully simplified

$Version

int=Integrate[((((2x+a+b))^2)/((((x+a))^2)(((x+b))^2)+(((2x+a+b))^2))),{x,-∞,∞}]

Timing[int2=FullSimplify[PowerExpand[int]]]

Timing[int3=FullSimplify[int2,a>0&&b>0]]

ran=Table[Random[],{10},{2}]

{#1,#2,int /.{a->#1,b->#2}}&@@@ran

{#1,#2,int2 /.{a->#1,b->#2}}&@@@ran