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Legendre's Chi-Function

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LegendresChi-Function

The function defined by

chi_nu(z)=sum_(k=0)^infty(z^(2k+1))/((2k+1)^nu).
(1)

It is related to the polylogarithm by

chi_nu(z)=1/2[Li_nu(z)-Li_nu(-z)]
(2)
=Li_nu(z)-2^(-nu)Li_nu(z^2)
(3)

and to the Lerch transcendent by

chi_nu(z)=2^(-nu)zPhi(z^2,nu,1/2).
(4)

It takes the special values

chi_2(i)=iK
(5)
chi_2(sqrt(2)-1)=1/(16)pi^2-1/4[ln(sqrt(2)+1)]^2
(6)
chi_2(1/2(sqrt(5)-1))=1/(12)pi^2-3/4[ln(1/2(sqrt(5)+1))]^2
(7)
chi_2(sqrt(5)-2)=1/(24)pi^2-3/4[ln(1/2(sqrt(5)+1))]^2
(8)
chi_2(-1)=-1/8pi^2
(9)
chi_2(1)=1/8pi^2,
(10)

where i is the imaginary unit and K is Catalan's constant (Lewin 1958, p. 19). Other special values include

chi_n(1)=lambda(n)
(11)
chi_n(i)=ibeta(n),
(12)

where lambda(n) is the Dirichlet lambda function and beta(n) is the Dirichlet beta function.

See also

Lerch Transcendent, Polylogarithm

Portions of this entry contributed by Joe Keane

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References

Cvijović, D. and Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205-210, 1995.Edwards, J. A Treatise on the Integral Calculus, Vol. 2. New York: Chelsea, p. 290, 1955.Legendre, A. M. Exercices de calcul intégral, tome 1. p. 247, 1811.Lewin, L. "Legendre's Chi-Function." §1.8 in Dilogarithms and Associated Functions. London: Macdonald, pp. 17-19, 1958.Lewin, L. Polylogarithms and Associated Functions. Amsterdam, Netherlands: North-Holland, pp. 282-283, 1981.Nielsen, N. "Der Eulersche Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh. der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch. 90, 121-212, 1909.

Referenced on Wolfram|Alpha

Legendre's Chi-Function

Cite this as:

Keane, Joe and Weisstein, Eric W. "Legendre's Chi-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LegendresChi-Function.html

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