Theta Function

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Related to Theta-function: Step function, Riemann theta function
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.)
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Here we have used the Fourier series representation of the theta-function (9),
Application of Jacobi's imiginary transformation [13, 6] to the two theta-functions occuring in (22) yields the result; the arising Jacobi theta-function [[upsilon].sub.2] is as in [6].
Key words and phrases : Gaussian function, Jacobi theta-functions, generator, dual function, interpolating function, interpolation theorem, Gauss transform, exponential Sobolev space, sampling theorem
Keywords and Phrases: Theta-functions, Modular equations, Eta-function identities.
If we set a = [qe.sup.2iz], b = [qe.sup.-2iz] and q = [e.sup.[pi]i[tau]], where z is complex and Im [tau] [greater than or equal to] 0, then f(a,b) = [[??].sub.3](z, [tau]), where [[??].sub.3](z, [tau]) denotes one of the classical theta-functions in its standrd notation[12].
Watson, Chapter 16 of Ramanujan's Second Notebook: Theta-functions and q-series, Mem.
The three special cases of Ramanujan's theta-function are as follows:
In Section 3, we establish several new modular equations of degree 2 for the ratios of Ramanujan's theta-function. In Section 4, we establish several general formulas for the explicit evaluations of [h.sub.2,n], for positive rational values of n.
Chandankumar, On some explicit evaluation of the ratios of Ramanujan's theta-function, (communicated).