Weber Functions

Author

Eric W. Weisstein
August 26, 2008

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

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

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

Notation

WeberE[ν,x]//TraditionalForm

Bessel-Type

v5.2

Integrate[Sin[ν θ-z Sin[θ]],{θ,0,π}]/π

v6

Integrate[Sin[ν θ-z Sin[θ]],{θ,0,π}]/π

V7

Integrate[Sin[ν θ-z Sin[θ]],{θ,0,π}]/π//InputForm

Integral

Table[(1/π)Integrate[Sin[ν θ-z Sin[θ]],{θ,0,π},Assumptions->z∈Reals],{ν,0,4}]

Hypergeometric

FunctionExpand[WeberE[ν,x]]//TraditionalForm

FullSimplify[%]

Total[FullSimplify[{((2 x HypergeometricPFQ[{1},{(3/2)-(ν/2),(3/2)+(ν/2)},-((x^2)/4)])/(π (-1+ν) (1+ν))),((2 HypergeometricPFQ[{1},{1-(ν/2),1+(ν/2)},-((x^2)/4)] )/(π ν))}]{((Cos[((π ν)/2)])^2),((Sin[((π ν)/2)])^2)}]

FullSimplify[%-FunctionExpand[WeberE[ν,x]]]

TraditionalForm[%8]

Table[WeberE[ν,x],{ν,0,4}]

TraditionalForm[%]

FunctionExpand

expr=FunctionExpand[WeberE[ν,x],ν∈Integers]

Table[expr,{ν,0,4}]

Table[((2 x ((Cos[((π ν)/2)])^2) HypergeometricPFQ[{1},{(3/2)-(ν/2),(3/2)+(ν/2)},-((x^2)/4)])/(π (-1+ν) (1+ν)))+((2 HypergeometricPFQ[{1},{1-(ν/2),1+(ν/2)},-((x^2)/4)] ((Sin[((π ν)/2)])^2))/(π ν)),{ν,0,4}]//FullSimplify

Even

FunctionExpand[WeberE[n,x],Mod[n,2]==0&&n>1]

FunctionExpand[WeberE[n,x],Element[n/2,Integers]&&n>1]

Table[%,{n,0,4,2}]

FullSimplify[%]

Odd

FunctionExpand[WeberE[n,x],Mod[n,2]==1&&n>2]

FunctionExpand[WeberE[n,x],Element[(n-1)/2,Integers]&&n>2]

Table[%,{n,1,6,2}]

FullSimplify[%]

Table

Table[(1/π)Integrate[Sin[ν θ-z Sin[θ]],{θ,0,π},Assumptions->z∈Reals],{ν,0,4}]//FullSimplify//Column//TraditionalForm

Elliptic-Type

f[z_]:=DedekindEta[(z+1)/2]/Exp[2Pi I/48]/DedekindEta[z]
f1[z_]:=DedekindEta[z/2]/DedekindEta[z]
f2[z_]:=Sqrt[2]DedekindEta[2z]/DedekindEta[z]
γ2[z_]:=(f[z]^24-16)/f[z]^8
γ3[z_]:=(f[z]^24+8)(f1[z]^8-f2[z]^8)/f[z]^8

Table[Subtract@@{f[z+1],f1[z]/Exp[2Pi I/48]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union

Table[Subtract@@{f1[z+1],f[z]/Exp[2Pi I/48]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union

Table[Subtract@@{f2[z+1],f2[z]Exp[2Pi I/24]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union

Table[Subtract@@{f[-1/z],f[z]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union

Table[Subtract@@{f1[-1/z],f2[z]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union

Table[Subtract@@{f2[-1/z],f1[z]}/.z->Random[Complex,{-2,2I},20],{100}]//Chop//Union