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Weierstrass Sigma Function

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WeierstrassSigmaReIm
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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/

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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.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 Resource. https://mathworld.wolfram.com/WeierstrassSigmaFunction.html

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