Theta Function


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Theta Function

 

Theta functions are entire functions whose quotients are elliptic functions.

The four principal theta functions are defined by the following rapidly converging series:

θ1(z) = 2q1/4 sin z – 2q9/4 sin 3z + 2q25/4 sin 5z – . . .

θ2(z) = 2q1/4 cos z – 2q9/4 cos 3z + 2q25/4 cos 5z + . . .

θ3(z) = 1 + 2q cos 2z + 2q4 cos 4z + 2q9 cos 6z + . . .

θ4(z) = 1 – 2q cos 2z + 2q4 cos 4z – 2q9 cos 6z + . . .

where ǀqǀ < 1.

When π is added to the argument z, the functions are multiplied by –1, –1, 1, and 1, respectively. When πτ is added to z, the relation between τ and q being given by the equation q = eπiτ the functions are multiplied by the factors – N, N, N, and –N, respectively, where N = q–1e–2ik. It follows that, for example, the quotient θ1(Z)/θ4(Z) is a meromorphic function that does not change when 2π or irr is added to the argument—that is, this quotient is an elliptic function with periods 2π and πτ.

For the representation of automorphic functions, H. Poincaré constructed theta functions that are a generalization of the theta functions described above, which were introduced by K. Jacobi (Jacobi’s notation was somewhat different).

REFERENCE

Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd edition, part 2. Moscow, 1963. (Translated from English.)
References in periodicals archive ?
The study of theta functions and theta constants has a long history, and they are very important objects in arithmetic and geometry.
Borwein, "Cubic analogues of the Jacobian theta function [theta](z;q)," Canadian Journal of Mathematics, vol.
Moreover by recasting these identities (27) and (30) in terms of theta function identity, we can easily discover and deduce other companion identities from the basic properties of the theta functions.
0]] (q) is a mock theta function, hence by mathematical induction are mock theta functions.
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The following formula [5,56] for the theta function holds:
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This information is obtained with the help of a relation that generalises the reciprocity law for Jacobi's theta functions (see Lemma 2).
SAFE, Convergence of Pade approximants of partial theta functions and the Rogers-Szego polynomials, Constr.
the partial mock theta function will be defined and denoted as
Miezaki, Congruences on the Fourier coefficients of the Mathieu mock theta function.
A Brief Introduction to Theta Functions (reprint, 1961)