Weierstrass Zeta Function

The Weierstrass zeta function  is the quasiperiodic function defined by

(1)

where  is the Weierstrass elliptic function with invariants  and , with

(2)

As in the case of other Weierstrass elliptic functions, the elliptic invariants  and  are frequently suppressed for compactness. The function is implemented in the Wolfram Language as WeierstrassZeta[u, g2, g3].

Using the definition above gives

			(3)
			(4)

where , so

(5)

so  is an odd function. Integrating  gives

(6)

Letting  gives

(7)

so

(8)

Similarly,

(9)

From Whittaker and Watson (1990),

(10)

If , then

(11)

(Whittaker and Watson 1990, p. 446). Also,

(12)

(Whittaker and Watson 1990, p. 446).

The series expansion of  is given by

(13)

where

			(14)
			(15)

and

(16)

for  (Abramowitz and Stegun 1972, p. 635). The first few coefficients are therefore

			(17)
			(18)
			(19)
			(20)
			(21)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Weierstrass Elliptic and Related Functions." Ch. 18 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 627-671, 1972.

Brezhnev, Y. V. "Uniformisation: On the Burnside Curve ." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.

Tölke, F. "Spezielle Weierstraßsche Zeta-Funktionen." Ch. 8 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 145-163, 1967.

Whittaker, E. T. and Watson, G. N. "Quasi-Periodic Functions. The Function " and "The Quasi-Periodicity of the Function ." §20.4 and 20.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 445-447 and 449-451, 1990.