Theta function

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.^[1]

For other θ functions, see Theta function (disambiguation).

The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called $z$ ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.

One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".^[2]

Throughout this article, $(e^{\pi i\tau })^{\alpha }$ should be interpreted as $e^{\alpha \pi i\tau }$ (in order to resolve issues of choice of branch).^{[note 1]}

Explicit values[edit]

Lemniscatic values[edit]

Proper credit for most of these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).^[4] Define,

Nome power theorems[edit]

Direct power theorems[edit]

For the transformation of the nome^[7] in the theta functions these formulas can be used:

Some series identities[edit]

Sums with theta function in the result[edit]

The infinite sum^[8]^[9] of the reciprocals of Fibonacci numbers with odd indices has this identity:

Relation to the Heisenberg group[edit]

The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. This invariance is presented in the article on the theta representation of the Heisenberg group.

Derivation of the theta values[edit]

Identity of the Euler beta function[edit]

In the following, three important theta function values are to be derived as examples:

This is how the Euler beta function is defined in its reduced form:

Partition sequences and Pochhammer products[edit]

Regular partition number sequence[edit]

The regular partition sequence $P(n)$ itself indicates the number of ways in which a positive integer number $n$ can be splitted into positive integer summands. For the numbers $n=1$ to $n=5$ , the associated partition numbers $P$ with all associated number partitions are listed in the following table:

; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. sec. 16.27ff. ISBN 978-0-486-61272-0.

Abramowitz, Milton

(1990) [1970]. Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs. Vol. 79. Providence, RI: AMS. ISBN 978-0-8218-4532-5.

Akhiezer, Naum Illyich

; Kra, Irwin (1980). Riemann Surfaces. New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8.. (for treatment of the Riemann theta)

Farkas, Hershel M.

; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford: Clarendon Press.

Hardy, G. H.

(1983). Tata Lectures on Theta I. Boston: Birkhauser. ISBN 978-3-7643-3109-2.

Mumford, David

(1959). Functions of a Complex Variable. New York: Dover Publications.

Pierpont, James

; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams & Wilkins. ISBN 978-0-683-07196-2.

Rauch, Harry E.

Reinhardt, William P.; Walker, Peter L. (2010), , in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

"Theta Functions"

; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's $θ$ functions)

Whittaker, E. T.

Farkas, Hershel M. (2008). "Theta functions in complex analysis and number theory". In (ed.). Surveys in Number Theory. Developments in Mathematics. Vol. 17. Springer-Verlag. pp. 57–87. ISBN 978-0-387-78509-7. Zbl 1206.11055.

Alladi, Krishnaswami

(1974). "IX. Theta series". Elliptic modular functions. Die Grundlehren der mathematischen Wissenschaften. Vol. 203. Springer-Verlag. pp. 203–226. ISBN 978-3-540-06382-7.

Schoeneberg, Bruno

Ackerman, Michael (1 February 1979). "On the generating functions of certain Eisenstein series". Mathematische Annalen. 244 (1): 75–81. :10.1007/BF01420339. S2CID 120045753.

doi

Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3.

Moiseev Igor. .

"Elliptic functions for Matlab and Octave"

This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.