# Sigma Functions

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## Sigma Functions

entire transcendental functions introduced by K. Weierstrass in his theory of elliptic functions. The principal sigma function (there are four in all) is

where w = 2mω_{1}+ 2nω_{2} (ω_{1} and ω_{2} are two numbers such that ω_{1}/ω_{2} is not real) and m and n independently run through all positive and negative integers, other than m = n = 0. The function σ(z) has simple zeros when *z* = *w*— that is, at the vertices of the parallelograms forming a regular lattice in the z-plane. These parallelograms are obtained from the parallelogram with vertices at the points 0, 2ω_{1}, 2ω_{2}, and 2(ω_{1} + ω_{2}) by means of translations along its sides.

The functionσ(*z) can be used to determine the Weierstrass zeta function ζ(z) and the Weierstrass elliptic function ℘(z):*

Let ω_{3} = – ω_{1} – ω_{2} and ζ(ω_{k}) = η_{K}, where k = 1, 2, 3. The formulas

σ(z + 2w_{k}) = – σ(z) exp [2η_{k}(z + w_{k})] k = 1, 2, 3

express the property of quasi periodicity of the function σ(z). The equations

define the three remaining sigma functions. We have σ(0) = 0 and σ_{k}(0) = 1, k = 1, 2, 3. The function σ(z) is odd, and the other sigma functions are even.

Any elliptic function *f*(z) with periods 2ω_{1}, and 2ω_{2} can be rationally expressed in terms of sigma functions by means of the formula

where *C* is a constant and *a*_{1} …, *a*, and *b*_{1}, …, *b*_{r} are, respectively, complete systems of zeros and poles of *f*(*z*) satisfying the condition *a*_{1} + … + *a _{r}* =

*b*

_{1}+ … +

*b*. Sigma functions are closely related to theta functions.

_{r}### REFERENCES

Smirnov, V. I.*Kurs vysshei matematiki*, 8th ed., vol. 3, part 2. Moscow, 1969.

Hurwitz, A., and R. Courant.

*Teoriia funktsii*. Moscow, 1968. (Translated from German.)

Whittaker, E. T., and G. N. Watson.

*Kurs sovremennogo analiza*, 2nd ed., part 2. Moscow, 1963. (Translated from English.)