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Weierstrass Zeta Function


WeierstrassZetaReImWeierstrassZetaContours

The Weierstrass zeta function zeta(z;g_2,g_3) is the quasiperiodic function defined by

 (dzeta(z;g_2,g_3))/(dz)=-P(z;g_2,g_3),
(1)

where P(z;g_2,g_3) is the Weierstrass elliptic function with invariants g_2 and g_3, with

 lim_(z->0)[zeta(z;g_2,g_3)-z^(-1)]=0.
(2)

As in the case of other Weierstrass elliptic functions, the elliptic invariants g_2 and g_3 are frequently suppressed for compactness. The function is implemented in the Wolfram Language as WeierstrassZeta[u, {g2, g3}].

Using the definition above gives

zeta(z)-z^(-1)=-int_0^z[P(z)-z^(-2)]dz
(3)
=-sum^'_(m,n=-infty)^inftyint_0^z[(z-Omega_(mn))^(-2)-Omega_(mn)^(-2)]dz,
(4)

where Omega_(mn)=2momega_1+2nomega_2, so

 zeta(z)=z^(-1)+sum^'_(m,n=-infty)^infty[(z-Omega_(mn))^(-1)+Omega_(mn)^(-1)+zOmega_(mn)^(-2)]
(5)

so zeta(z) is an odd function. Integrating P(z+2omega_1)=P(z) gives

 zeta(z+2omega_1)=zeta(z)+2eta_1.
(6)

Letting z=-omega_1 gives

 zeta(-omega_1)+2eta_1=-zeta(omega_1)+2eta_1,
(7)

so

 eta_1=zeta(omega_1).
(8)

Similarly,

 eta_2=zeta(omega_2).
(9)

From Whittaker and Watson (1990),

 eta_1omega_2-eta_2omega_1=1/2pii.
(10)

If x+y+z=0, then

 [zeta(x)+zeta(y)+zeta(z)]^2+zeta^'(x)+zeta^'(y)+zeta^'(z)=0
(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 zeta(z) is given by

 zeta(z)=z^(-1)-sum_(k=2)^infty(c_kz^(2k-1))/(2k-1),
(13)

where

c_2=(g_2)/(20)
(14)
c_3=(g_3)/(28)
(15)

and

 c_k=3/((2k+1)(k-3))sum_(m=2)^(k-2)c_mc_(k-m)
(16)

for k>=4 (Abramowitz and Stegun 1972, p. 635). The first few coefficients are therefore

c_4=1/3c_2^2
(17)
c_5=3/(11)c_2c_3
(18)
c_6=1/(39)(2c_2^3+3c_3^2)
(19)
c_7=2/(33)c_2^2c_3
(20)
c_8=5/(7293)(11c_2^4+36c_3^2c_2).
(21)

See also

Weierstrass Elliptic Function, Weierstrass Sigma Function

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 y^2=x^5-x." 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 zeta(z)" and "The Quasi-Periodicity of the Function zeta(z)." §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.

Referenced on Wolfram|Alpha

Weierstrass Zeta Function

Cite this as:

Weisstein, Eric W. "Weierstrass Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassZetaFunction.html

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