TOPICS
Search

Abel's Functional Equation

Let L(x) denote the Rogers L-function defined in terms of the usual dilogarithm by

L(x)=6/(pi^2)[Li_2(x)+1/2lnxln(1-x)]
(1)
=6/(pi^2)[sum_(n=1)^(infty)(x^n)/(n^2)+1/2lnxln(1-x)],
(2)

then L(x) satisfies the functional equation

L(x)+L(y)=L(xy)+L((x(1-y))/(1-xy))+L((y(1-x))/(1-xy)).
(3)

Abel's duplication formula follows from this identity.

See also

Abel's Duplication Formula, Dilogarithm, Functional Equation, Polylogarithm, Riemann Zeta Function, Rogers L-Function

Explore with Wolfram|Alpha

References

Abel, N. H. Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 189-192, 1988.Bytsko, A. G. "Two-Term Dilogarithm Identities Related to Conformal Field Theory." 9 Nov 1999. http://arxiv.org/abs/math-ph/9911012.Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431-448, 1997.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 14 and 21, 1999.Rogers, L. J. "On Function Sum Theorems Connected with the Series sum_1^(infty)x^n/n^2." Proc. London Math. Soc. 4, 169-189, 1907.

Referenced on Wolfram|Alpha

Abel's Functional Equation

Cite this as:

Weisstein, Eric W. "Abel's Functional Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelsFunctionalEquation.html

Subject classifications