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Jonquière's Relation

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Jonquière's relation, sometimes also spelled "Joncquière's relation" (Erdélyi et al. 1981, p. 31), states

Li_s(z)+e^(piis)Li_s(1/z)=((2pi)^se^(ipis/2))/(Gamma(s))zeta(1-s,(lnz)/(2pii))

Erdélyi et al. (1981, p. 31), where Li_s(z) is a polylogarithm, Gamma(s) is the gamma function, and zeta(s,w) is the Hurwitz zeta function, and z is not a member of the real interval [0,1].

The most general form of the identity valid everywhere in the complex plane is

Li_s(z)=(ipi)/(Gamma(s))(1-sqrt((z-1)/z)sqrt(z/(z-1)))ln^(s-1)z+(e^(ipis/2)(2pi)^s)/(Gamma(s))zeta(1-s,(lnz)/(2pii))-e^(ipis)Li_s(1/z).

See also

Polylogarithm

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 31, 1981.Jonquière, A. "Note sur la série sum_(n=1)^(n=infty)(x^n)/(n^s)." Bull. Soc. Math. France 17, 142-152, 1889.Sondow, J. and Hadjicostas, P. "The Generalized-Euler-Constant Function gamma(z) and a Generalization of Somos's Quadratic Recurrence Constant." 16 Oct 2006. http://arxiv.org/abs/math.CA/0610499.

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Jonquière's Relation

Cite this as:

Weisstein, Eric W. "Jonquière's Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JonquieresRelation.html

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