Van der Corput's Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/vanderCorputsConstant.nb.

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

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

Value

FullSimplify[Integrate[Sin[x^2+c],{x,0,(Sqrt[(π/2)-c])}],-π/2<c<π/2]

(Sqrt[(π/2)]) (C((Sqrt[1-((2 c)/π)])) sin(c)+S((Sqrt[1-((2 c)/π)])) cos(c))

FindInstance[(Sqrt[(π/2)]) (Cos[c] FresnelS[(Sqrt[1-((2 c)/π)])]+FresnelC[(Sqrt[1-((2 c)/π)])] Sin[c])==0,c]

2(Sqrt[2])Integrate[Cos[(x^2)+c],{x,0,(Sqrt[(π/2)-c])}]//FullSimplify

TraditionalForm[vdc=With[{c=Root[{Cos[#1] FresnelS[(Sqrt[1-((2 #1)/π)])]+FresnelC[(Sqrt[1-((2 #1)/π)])] Sin[#1]&,-0.726643246813242509590219555941985563257278933469994008688`30.}]},2 (Sqrt[π]) (Cos[c] FresnelC[(Sqrt[1-((2 c)/π)])]-FresnelS[(Sqrt[1-((2 c)/π)])] Sin[c])]]

N[vdc,1000]

InputForm[ee=Internal`ToStraightLineProgram[vdc/.Root->r]/.r->Root]

Internal`FromStraightLineProgram[ee]