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# Weierstrass Sigma Function

The quasiperiodic function defined by

 (1)

where is the Weierstrass zeta function and

 (2)

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

 (3)

where the term with is omitted from the product and .

Amazingly, , where is the Weierstrass sigma function with half-periods and , has a closed form in terms of , , and . This constant is known as the Weierstrass constant.

 (4) (5)

and

 (6)

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

can be expressed in terms of Jacobi theta functions using the expression

 (7)

where , and

 (8) (9)

There is a beautiful series expansion for , given by the double series

 (10)

where , for either subscript negative, and other values are gives by the recurrence relation

 (11)

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

 1 -3 -54 14904 -1 -18 4968 502200 -9 513 257580 162100440 69 33588 20019960 -9465715080 321 2808945 -376375410 -4582619446320 160839 -41843142 -210469286736 -1028311276281264

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 ." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.Knopp, K. "Example: Weierstrass's -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 ." §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