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 = 2mω1+ 2nω21 and ω2 are two numbers such that ω12 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, 2ω1, 2ω2, and 2(ω1 + ω2) 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 ω3 = – ω1 – ω2 and ζ(ωk) = ηK, where k = 1, 2, 3. The formulas

σ(z + 2wk) = – σ(z) exp [2ηk(z + wk)] 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 σk(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 2ω1, and 2ω2 can be rationally expressed in terms of sigma functions by means of the formula

where C is a constant and a1 …, a, and b1, …, br are, respectively, complete systems of zeros and poles of f(z) satisfying the condition a1 + … + ar = b1 + … + br. Sigma functions are closely related to theta functions.

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.)
References in periodicals archive ?
Besides, he also studied the mean value problem of exponential divisor function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and obtained:
Zhai, On the Smarandache ceil function and the Dirichlet divisor function, Scientia Magna, 4(2008), No.
In [3], Gou Su studied the hybrid mean value of Smarandache kn sequence and divisor function [sigma](n), and gave the following theorem:
Suppose that 1 [less than or equal to] a [less than or equal to] b are fixed integers, the divisor function d(a, b; k) is defined by
For fixed integers 1 [less than or equal to] a [less than or equal to] b, the divisor function d(a, b; n) is defined by
where d(n) is the Dirichlet divisor function and [B.
We shall study the mean value of the function (t(e))r, which is closely related to the general divisor function [d.
e)](n) denote the number of exponential divisors of n, which is called the exponential divisor function.
In order to prove our theorem, we define for an arbitrary complex number z the general divisor function [d.
d = n 2, where d(n) is the Dirichlet divisor function.
3] (n) denote the Piltz divisor function of dimensional 3, then for any real number x [greater than or equal to] 2, we have