Schlömilch's Series

Author

Eric W. Weisstein
August 19, 2008

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

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

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

Code

SchloemilchA[f_,x_,0]:=Module[{u,phi},
2Function[x,f][0]+Integrate[u Derivative[1][Function[x,f]][u Sin[phi]],{u,0,Pi},{phi,0,Pi/2}]2/Pi
]

SchloemilchA[f_,x_,n_]:=Module[{u,phi},
Integrate[u Derivative[1][Function[x,f]][u Sin[phi]]Cos[n u],{u,0,Pi},{phi,0,Pi/2},Assumptions->n∈Integers&&n>0]2/Pi
]

Examples

x

SchloemilchA[x,x,0]

SchloemilchA[x,x,n]

V6

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

Sum[BesselJ[0,(2n-1) x]/(2n-1)^2,{n,∞}]

V7

((π^2)/4)+Sum[((-1+(((-1))^n))/(n^2))BesselJ[0,n x],{n,∞}]

$Version

(1/4)(π^2)-2Sum[((BesselJ[0,n x])/(n^2)),{n,1,∞,2},Assumptions->0<x<π]

Sum[BesselJ[0,(2n-1) x]/(2n-1)^2,{n,∞}]

Plot

Show[
Block[{$DisplayFunction=Identity},
{
Plot[x,{x,-1,4}],
ListPlot[Table[{x,
Pi^2/4+Sum[((-1+(((-1))^n))/(n^2))BesselJ[0,n x],{n,50}]
},{x,-1,4,.1}],Joined->True,PlotStyle->Red]
}],Epilog->{Dashing[{.02}],Blue,Line[{{Pi,0},{Pi,4}}]}]

Show[
Block[{$DisplayFunction=Identity},
{
Plot[x,{x,0,3.5}],
Sequence@@(ListPlot[Table[{x,
Pi^2/4+Sum[((-1+(((-1))^n))/(n^2))BesselJ[0,n x],{n,2#-1}]
},{x,0,3.5,.1}],Joined->True,PlotStyle->{Red,Green,Blue,Violet}[[#]]]&/@Range[4])
}],Epilog->{Dashing[{.02}],Blue,Line[{{Pi,0},{Pi,4}}]},
AxesLabel->TraditionalForm/@{x,f[x]}]

(x^2)

SchloemilchA[(x^2),x,0]

SchloemilchA[(x^2),x,n]

Show[
Block[{$DisplayFunction=Identity},
{
Plot[x^2,{x,-4,4}],
ListPlot[Table[{x,
((2 (π^2))/3)+Sum[((8(((-1))^n))/(n^2))BesselJ[0,n x],{n,50}]
},{x,-4,4,.1}],Joined->True,PlotStyle->Red]
}],Epilog->{Dashing[{.02}],Blue,Line[{{Pi,0},{Pi,20}}]}]

V7

((2(π^2))/3)+8Sum[((((-1))^n)/(n^2))BesselJ[0,n x],{n,∞}]

Subtract@@@Table[{(x^2),((2(π^2))/3)+8Sum[((((-1))^n)/(n^2))BesselJ[0,n x],{n,5 10^4}]}/.x->RandomReal[π{-1,1}],{10}]//Chop//Union

(x^p)

Table[SchloemilchA[(x^p),x,0],{p,10}]

Table[SchloemilchA[(x^p),x,n],{p,5}]

Table[%,{n,5}]

cos(x)

SchloemilchA[Cos[x],x,0]

SchloemilchA[Cos[x],x,n]

sin(x)

SchloemilchA[Sin[x],x,0]

SchloemilchA[Sin[x],x,n]

Integrate[u BesselJ[0,u]Cos[n u],{u,0,Pi}]

Table[Integrate[u BesselJ[0,u]Cos[n u],{u,0,Pi}],{n,2}]

FunctionExpand[(1/2) (π^2) HypergeometricPFQ[{(1/4),(3/4)},{(1/2),(1/2),2},-(π^2)]]

(e^x)

SchloemilchA[Exp[x],x,0]

SchloemilchA[Exp[x],x,n]

(Subscript[J, 0])(x)

V7

SchloemilchA[BesselJ[0,x],x,0]//Timing

SchloemilchA[BesselJ[0,x],x,n]//Timing