Jacobi-Anger Expansion

Author

Eric W. Weisstein
September 21, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Jacobi-AngerExpansion.nb.

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

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

Sums

(∑_(n=-∞)^∞)(i^n) exp(i n θ) (Subscript[J, n])(z)==exp(i z cos(θ))

Sum[(I^n)BesselJ[n,z]Exp[I n θ],{n,-∞,∞}]

{Sum[(I^n)BesselJ[n,z]Exp[I n θ],{n,-10,10}],Exp[I z Cos[θ]]}/.{z->RandomComplex[],θ->RandomReal[{0,2π}]}

(Subscript[J, 0])(z)+2 (∑_(n=1)^∞)(i^n) (Subscript[J, n])(z) cos(n θ)==exp (i z θ cos)

BesselJ[0,z]+2Sum[(I^n)BesselJ[n,z]Cos[n θ],{n,1,∞}]

{BesselJ[0,z]+2Sum[(I^n)BesselJ[n,z]Cos[n θ],{n,1,10}],Exp[I z Cos[θ]]}/.{z->RandomComplex[],θ->RandomReal[{0,2π}]}