Mehler's Bessel Function Formula

Author

Eric W. Weisstein
November 29, 2008

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

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

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

Formula

(2/π)Integrate[Sin[x Cosh[t]],{t,0,∞}]

Numerical Check

With[{x=.01},Plot[Sin[x Cosh[t]],{t,0,10}]]

With[{x=1.},Plot[Sin[x Cosh[t]],{t,0,10}]]

BesselJ[0,.01]

With[{x=.01},(2/π)NIntegrate[Sin[x Cosh[t]],{t,0,∞}]]

With[{x=.01},(2/π)NIntegrate[Sin[x Cosh[t]],{t,0,∞},Method->"Oscillatory"]]

For Sin[f[x]], f non-rapidly-oscillatory, you can use LevinIntegrate which applies Levin' s method.Levin' s method will be in NIntegrate soon:

Needs["Moylan`"]

With[{x=.01},2/π LevinIntegrate[Sin[x Cosh[t]],{t,0,∞}]]