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.

In addition, satisfies

			(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

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