Struve Function

Author

Eric W. Weisstein
December 30, 2008

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

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

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

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

Plot[Evaluate[Table[StruveH[n,z],{n,0,5}]],{z,-10,10},PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},AxesLabel->TraditionalForm/@{x,StruveH[n,x]}]

Complex Plane

ComplexPlot3D[StruveH[0,z],{z,-5,5}]//Timing

ComplexContourPlot[StruveH[0,z],{z,-5,5}]//Timing

Special Values

FunctionExpand[StruveH[0,z]]

FunctionExpand[StruveH[1,z]]

FullSimplify[Table[FunctionExpand[StruveH[n/2,z]],{n,1,5,2}]]

Series

(Subscript[H, 0])(z)

FullSimplify[SeriesCoefficient[StruveH[0,z]Pi/2,{z,0,n}],n∈Integers&&n>0]

FullSimplify[-((((((I/2)))^(1+n)) π)/((Gamma[1+(n/2)])^2)),Mod[n,2]==1&&n>0]

Table[-((((((I/2)))^(1+n)) π)/((Gamma[1+(n/2)])^2)),{n,1,20,2}]

FullSimplify[FindSequenceFunction[%,n],n∈Integers]

s=Series[StruveH[0,z]Pi/2,{z,0,20}]

Through[{Numerator,Denominator}[CoefficientList[Normal[s]/z,z^2]]]

Table[((2n-1)!!)^2,{n,10}]

Table[((((-1))^n)/(((2n+1)!!)^2))(z^(2n+1)),{n,0,6}]

(Subscript[H, 1])(z)

s=Series[StruveH[1,z]Pi/2,{z,0,20}]

Through[{Numerator,Denominator}[CoefficientList[Normal[s],z^2]]]

Table[(2n-1)!!(2n+1)!!,{n,0,10}]

Sum[((((-1))^(n+1))/((2n-1)!!(2n+1)!!))(z^(2n)),{n,∞}]

(Subscript[H, 3])(z)

s=Series[StruveH[3,z]Pi/2,{z,0,20}]

Through[{Numerator,Denominator}[CoefficientList[Normal[s],z^2]]]

Approximation for (Subscript[H, 1])(z)

Series[StruveH[1,z],{z,0,4}]

h[z_?NumericQ]:=(2/π)-BesselJ[0,z]+((16/π)-5)((Sin[z])/z)+(12-(36/π))((1-Cos[z])/(z^2))

<<Utilities`Plot`

Plot[StruveH[1,x]-h[x],{x,-50,50},PlotPoints->100,PlotStyle->Red,AxesLabel->TraditionalForm/@{x,StruveH[1,x]-h[x]}]

ComplexPlot3D[StruveH[1,z]-g[z],{z,-2,2}]//Timing

ComplexPlot3D[StruveH[1,z]-g[z],{z,-10,10}]//Timing

ComplexContourPlot[StruveH[1,z]-g[z],{z,-2,2}]//Timing

ComplexContourPlot[StruveH[1,z]-g[z],{z,-10,10}]//Timing

Sums

((((z/2)))^(ν+1))(∑_(k=0)^∞)(((((-1))^k) ((((z/2)))^(2 k)))/((Gamma[k+(3/2)]) (Gamma[k+ν+(3/2)])))==(StruveH[ν, z])

V6

(((z/2))^(ν+1))Sum[(((((-1))^k)(((z/2))^(2k)))/(Gamma[k+3/2]Gamma[k+ν+3/2])),{k,0,∞}]

V7

(((z/2))^(ν+1))Sum[(((((-1))^k)(((z/2))^(2k)))/(Gamma[k+3/2]Gamma[k+ν+3/2])),{k,0,∞}]

(2/π) (∑_(n=0)^∞)(((((-1))^n) (z^(2 n+1)))/((((2 n+1)!!))^2))==(StruveH[0, z])

V6

Sum[((((-1))^n)/(((2n+1)!!)^2))(z^(2n+1)),{n,0,∞}]

V7

(2/π)Sum[((((-1))^n)/(((2n+1)!!)^2))(z^(2n+1)),{n,0,∞}]

(1/π)(∑_(k=0)^n)(((Gamma[k+(1/2)]) ((((z/2)))^(-2 k+n-1/2)))/(Gamma[-k+n+1]))==(StruveH[n+1/2, z])-(BesselY[n+1/2, z])

(1/π)Sum[((Gamma[k+(1/2)]((((z/2)))^(-2k+n-1/2)))/(Gamma[n+1-k])),{k,0,n}]//FullSimplify

Subtract@@@Table[{(1/π)Sum[((Gamma[k+(1/2)]((((z/2)))^(-2k+n-1/2)))/(Gamma[n+1-k])),{k,0,n}],StruveH[n+1/2,z]-BesselY[n+1/2,z]},{n,5}]

FullSimplify[StruveH[n+1/2,z]-BesselY[n+1/2,z],n∈Integers&&n>0]

2 (∑_n^∞)-(((((-(1/4)))^n) (x^(2 n-1)))/((Gamma[n+(1/2)])^2))==(StruveH[0, x])

2Sum[-((((-(1/4)))^n)/((Gamma[(1/2)+n])^2))(x^(2n-1)),{n,∞}]

((2 )/π)(∑_k^∞)(((((-1))^(k+1)) (z^(2 k)))/((2 k-1)!! (2 k+1)!!))==(StruveH[1, z])

(2/π)Sum[((((-1))^(k+1))/((2k-1)!!(2k+1)!!))(z^(2k)),{k,∞}]

Generating Functions

(StruveH[0, z])

SeriesCoefficient[StruveH[0,z],{z,0,n}]//FullSimplify

(StruveH[1, z])

SeriesCoefficient[StruveH[1,z],{z,0,n}]//FullSimplify

Integrals

((2 ((((z/2)))^ν))/((Gamma[(1/2)]) (Gamma[ν+(1/2)])))(∫_0^1)(((1-(t^2)))^(ν-(1/2))) sin(t z)dt==(StruveH[ν, z])

V6

((2(((z/2))^ν))/(Gamma[ν+1/2]Gamma[1/2]))Integrate[(((1-(t^2)))^(ν-1/2))Sin[z t],{t,0,1}]

((2(((z/2))^ν))/(Gamma[ν+1/2]Gamma[1/2]))Integrate[(((1-(t^2)))^(ν-1/2))Sin[z t],{t,0,1},GenerateConditions->False]

FullSimplify[(z^ν) ((((z^2)))^(-ν/2)) Sign[z] StruveH[ν,z Sign[z]],z∈Reals]

V7

((2(((z/2))^ν))/(Gamma[ν+1/2]Gamma[1/2]))Integrate[(((1-(t^2)))^(ν-1/2))Sin[z t],{t,0,1}]

((2(((z/2))^ν))/(Gamma[ν+1/2]Gamma[1/2]))Integrate[(((1-(t^2)))^(ν-1/2))Sin[z t],{t,0,1},GenerateConditions->False]

Identities

(StruveH[ν-1, z])-(StruveH[ν+1, z])==2 Derivative[0,1][StruveH](ν,z)-(((2^(-ν)) (z^ν))/((Sqrt[π]) (Gamma[ν+(3/2)])))

StruveH[ν-1,z]-StruveH[ν+1,z]-2Derivative[0,1][StruveH][ν,z]

FullSimplify[StruveH[ν-1,z]-StruveH[ν+1,z]]

Recurrences

RSolve[{H[ν-1,z]-H[ν+1,z]==2Derivative[0,1][StruveH][ν,z]-(((2^(-ν)) (z^ν))/((Sqrt[π]) Gamma[(3/2)+ν])),H[0,z]==StruveH[0,z],H[1,z]==StruveH[1,z]},H[ν,z],ν]//Timing

ODE

DSolve[z y''[z]+z y'[z]+((z^2)-(ν^2))y[z]==(2/π)((z^(n+1))/((2n-1)!!)),y[z],z]