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van der Corput's Constant

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Van der Corput's constant is given by

m=2sqrt(2)int_0^(sqrt(pi/2-c))cos(x^2+c)dx
(1)
=2sqrt(pi)[coscC(phi)-sincS(phi)]
(2)
=3.3643175781...
(3)

(OEIS A143305), where C(x) and S(x) are Fresnel integrals,

phi=sqrt(1-(2c)/pi),
(4)

and c is the unique negative root of

int_0^(sqrt(pi/2-c))sin(x^2+c)dx=sqrt(pi/2)[sincC(phi)+coscS(phi)]
(5)
=0
(6)

with -pi/2<c<0, namely

c=-0.726643246...
(7)

(Finch 2003, p. 245; OEIS A143306).

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References

Finch, S. R. "Van der Corput's Constant." §3.15 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 245-247, 2003.Sloane, N. J. A. Sequences A143305 and A143306 in "The On-Line Encyclopedia of Integer Sequences."

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van der Corput's Constant

Cite this as:

Weisstein, Eric W. "van der Corput's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/vanderCorputsConstant.html

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