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Weierstrass Sigma Function


WeierstrassSigmaReImWeierstrassSigmaContours

The quasiperiodic function defined by

 d/(dz)lnsigma(z;g_2,g_3)=zeta(z;g_2,g_3),
(1)

where zeta(z;g_2,g_3) is the Weierstrass zeta function and

 lim_(z->0)(sigma(z))/z=1.
(2)

(As in the case of other Weierstrass elliptic functions, the invariants g_2 and g_3 are frequently suppressed for compactness.) Then

 sigma(z)=zproduct_(m,n=-infty)^infty^'[(1-z/(Omega_(mn)))exp(z/(Omega_(mn))+(z^2)/(2Omega_(mn)^2))],
(3)

where the term with m=n=0 is omitted from the product and Omega_(mn)=2momega_1+2nomega_2.

Amazingly, sigma(1|1,i)/2, where sigma(z|omega_1,omega_2) is the Weierstrass sigma function with half-periods omega_1 and omega_2, has a closed form in terms of pi, e, and Gamma(1/4). This constant is known as the Weierstrass constant.

In addition, sigma(z) satisfies

sigma(z+2omega_1)=-e^(2eta_1(z+omega_1))sigma(z)
(4)
sigma(z+2omega_2)=-e^(2eta_2(z+omega_2))sigma(z)
(5)

and

 sigma_r(z)=(e^(-eta_rz)sigma(z+omega_r))/(sigma(omega_r))
(6)

for r=1, 2, 3. The function is implemented in the Wolfram Language as WeierstrassSigma[u, {g2, g3}].

sigma(z) can be expressed in terms of Jacobi theta functions using the expression

 sigma(z|omega_1,omega_2)=(2omega_1)/(pitheta_1^')exp(-(nu^2theta_1^('''))/(6theta_1^'))theta_1(nu|(omega_2)/(omega_1)),
(7)

where nu=piz/(2omega_1), and

eta_1=-(pi^2theta_1^('''))/(12omega_1theta_1^')
(8)
eta_2=-(pi^2omega_2theta_1^('''))/(12omega_1^2theta_1^')-(pii)/(2omega_1).
(9)

There is a beautiful series expansion for sigma(z), given by the double series

 sigma(z)=sum_(m,n=0)^inftya_(mn)(1/2g_2)^m(2g_3)^n(z^(4m+6n+1))/((4m+6n+1)!),
(10)

where a_(00)=1, a_(mn)=0 for either subscript negative, and other values are gives by the recurrence relation

 a_(mn)=3(m+1)a_(m+1,n+1)+(16)/3(n+1)a_(m-2,n+1) 
 -1/3(2m+3n-1)(4m+6n-1)a_(m-1,n)
(11)

(Abramowitz and Stegun 1972, pp. 635-636). The following table gives the values of the a_(mn) coefficients for small m and n.

n=0n=1n=2n=3
a_(0n)1-3-5414904
a_(1n)-1-184968502200
a_(2n)-9513257580162100440
a_(3n)693358820019960-9465715080
a_(4n)3212808945-376375410-4582619446320
a_(5n)160839-41843142-210469286736-1028311276281264

See also

Weierstrass Constant, Weierstrass Elliptic Function, Weierstrass Zeta Function

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/WeierstrassSigma/, http://functions.wolfram.com/EllipticFunctions/WeierstrassSigma4/

Explore with Wolfram|Alpha

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.Knopp, K. "Example: Weierstrass's sigma-Function." §2d in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 27-30, 1996.Tölke, F. "Spezielle Weierstraßsche Sigma-Funktionen." Ch. 9 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 164-180, 1967.Whittaker, E. T. and Watson, G. N. "The Function sigma(z)." §20.42 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 447-448, 450-452, and 458-461, 1990.

Referenced on Wolfram|Alpha

Weierstrass Sigma Function

Cite this as:

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

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