Glasser Function

Author

Eric W. Weisstein
April 24, 2002

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Graphics`Colors`;

Definition

Integrate[Sin[x Sin[x]],x]

Numerical Solution

Glasser=y/.NDSolve[{y'[x]==Sin[x Sin[x]],y[0]==0},y,{x,0,100},MaxSteps->10^5][[1]]

Plot[Glasser[x],{x,0,50},PlotPoints->1000,
AxesLabel->TraditionalForm/@{x,G[x]}]

Asymptotics

Plot[{2Sqrt[x/Pi],Glasser[x]},{x,0,50},PlotPoints->1000,
AxesLabel->TraditionalForm/@{x,G[x]},
PlotStyle->{Red,Black}]

Glasser's approximation

Doesn't seem right;also defined only for n>0

S[n π]~2Sqrt[n]-3/4π Sqrt[n]+(-1)^(n-1)/π n

S[n π+x]~S[n π]-Cos[(n π+x)Sin[x]]/n π Cos[n π+x]+O[1/n^2]

GInt[y_]:=Module[{n=Floor[y/Pi]},
2Sqrt[n]-3/(4π Sqrt[n])+(-1)^(n-1)/(π n)
]

Plot[{GInt[x],Glasser[x]},{x,Pi,50},PlotPoints->50,
AxesLabel->TraditionalForm/@{x,HoldForm[G[x]]},
PlotStyle->{Red,Black}]

G[y_]:=Module[{n=Floor[y/Pi],x},
x=y-n Pi;
2Sqrt[n]-3/(4π Sqrt[n])+(-1)^(n-1)/(π n)-Cos[(n π+x)Sin[x]]/(n π Cos[n π+x])
]

Plot[{G[x],Glasser[x]},{x,Pi,50},PlotPoints->50,
AxesLabel->TraditionalForm/@{x,HoldForm[G[x]]},
PlotStyle->{Red,Black}]

Plot of Derivatives

derivs=NestList[FullSimplify[D[#,x]]&,Sin[x Sin[x]],2]

Show[GraphicsArray[Plot[#1,{x,0,20},
PlotPoints->500,
AxesLabel->TraditionalForm/@{x,#2},
DisplayFunction->Identity]&@@@Transpose[{Rest[derivs],{G'[x],G''[x]}}]
]]

Similarity to Hundred-dollar problem

(∫_0^∞)Cos[x Exp[x]]\[DifferentialD]x