Weierstrass Zeta Function

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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)

(8)

Similarly,

(9)

From Whittaker and Watson (1990),

(10)

If , then

(11)

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

2(|1 P(x) P^2(x); 1 P(y) P^2(y); 1 P(z) P^2(z)|)/(|1 P(x) P^'(x); 1 P(y) P^'(y); 1 P(z) P^'(z)|)=zeta(x+y+z)-zeta(x)-zeta(y)-zeta(z)

(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)

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/WeierstrassZeta/, http://functions.wolfram.com/EllipticFunctions/WeierstrassZetaHalfPeriodValues/

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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