In function theory , a branch of mathematics , the theta functions form a special class of functions of several complex variables. They were first systematically examined by Carl Gustav Jakob Jacobi .
Theta functions play a role in the theory of elliptic functions and quadratic forms . They were introduced in 1829 by Jacobi in his book Fundamenta nova theoriae functionum ellipticarum . Jacobi used the Greek letter for her and gave her the name Theta function. For Jacobi it is the basis of his treatment of elliptical functions, systematically developed in his lectures. Carl Friedrich Gauß recognized the importance of the theta function for the theory of elliptical functions, but did not publish it. The theta function itself was already known to Leonhard Euler and Johann I Bernoulli in special cases . Further contributions to the theory of the theta function came in the 19th century in particular from Karl Weierstrass , Bernhard Riemann , Frobenius and Henri Poincaré .
Theta functions appear, for example, when solving the heat conduction equation .
definition
Classic theta function
The classic Jacobean theta function is defined by
The series is convergent in normal , where the upper half plane means . So for solid is a whole function , for solid is a holomorphic function .
More theta functions
In addition to the classic theta function, there are three other theta functions in the literature, namely:
The Jacobian theta function is referred to as
or in this notation .
One defines something more general
Theta zero
The theta zero value is understood to mean the theta function for the value
, for example the series for the Jacobian theta function
properties
zeropoint
For solid , the theta function has simple zeros at the points
-
.
Transformation formula
The theta function is periodic in both variables, it is
In addition, the important transformation formula applies
This is especially reduced for the theta zero value
The main branch is to be taken from the root.
Product presentation
The theta function can also be represented as an infinite product with the help of the Jacobian triple product, the following applies:
This is especially reduced for the theta zero value
From this representation it follows in particular that there are no zeros in the upper half-plane .
Integral representation
The theta function has an integral representation:
Differential equation
The theta function also plays an important role in the theory of heat conduction , for real ones and it is a solution of the partial differential equation
how to by inserting
sees. This corresponds to a Fourier expansion in spatial space with coefficients with an exponentially decreasing time dependence.
Jacobi identity
The theta zero values fulfill the so-called Jacobi identity:
Connection with the Riemann zeta function
In his famous work on the number of prime numbers under a given quantity, Riemann used the transformation formula of the theta function for a proof of the functional equation of the zeta function , namely:
Relationship with modular forms and elliptical functions
Connection with the Dedekind eta function
The theta function is closely related to the Dedekind eta function , the following applies:
The theta function as a module form to a subgroup of the module group
Module forms can be defined using the theta function. If one sets , then due to the transformation behavior
applies
The function is therefore a modular form of weight 4 to the subgroup of the
module group generated by the two transformations and .
Quotients of theta functions
The theta function can be used to define elliptic functions
. If one sets for solid :
-
,
so is an elliptic function to the lattice .
Weierstraß's ℘-function can be constructed in a similar way. If a holomorphic function fulfills the two conditions
and for a fixed
one, the second logarithmic derivative is an elliptic function to the lattice . For example, the Weierstrasse ℘ function applies:
with a matching constant .
Connection with number theoretic functions
With the help of the theta function and its product representation, the pentagonal number theorem can be
proven.
Another application is a formula for the third power of the Euler product:
See also
literature
-
Adolf Krazer : Textbook of the theta functions . BG Teubner, Leipzig 1903.
-
Milton Abramowitz , Irene Stegun : Handbook of Mathematical Functions . Dover, New York 1972; P. 576 .
-
Tom M. Apostol : Modular Functions and Dirichlet Series in Number Theory . Springer-Verlag, New York 1990, ISBN 0-387-97127-0
-
Adolf Hurwitz : Lectures on general function theory and elliptic functions . Springer-Verlag, Berlin / Heidelberg / New York 2000, ISBN 3-540-63783-4
-
Dale Husemöller : Elliptic Curves . Springer Verlag, Berlin / Heidelberg / New York 2004, ISBN 0-387-95490-2
-
Max Koecher and Aloys Krieg : Elliptical functions and modular forms . 2nd Edition. Springer-Verlag, Berlin / Heidelberg / New York 2007, ISBN 3-540-63744-3
-
Reinhold Remmert : Function theory I. Springer-Verlag Berlin Heidelberg New York 1989, ISBN 3-540-51238-1
-
David Mumford Tata Lectures on Theta . Volume 1. 3rd edition. Springer Verlag 1994 (three volumes in total)
-
Jun-Ichi Igusa Theta Functions . Basics of mathematical Sciences. Springer Verlag, 1972
-
Bruno Schoeneberg Elliptic Modular Functions . Basics of mathematical Sciences. Springer Verlag, 1974 (Chapter 9, Theta Series )
-
Harry Rauch , Hershel Farkas Theta Functions with Applications to Riemann Surfaces . Williams & Wilkins, Baltimore 1974
- Farkas, Irwin Kra Riemann Surfaces . Springer Verlag, Graduate Texts in Mathematics, 1980 (Chapter 6)
Web links
Individual evidence
-
^ Theory of elliptical functions derived from the properties of the theta series, elaboration of the lecture by Karl Wilhelm Borchardt 1838. In: Jacobi: Werke , Volume 1, 1881 (editor Borchardt, Karl Weierstrass ), pp. 497-538
-
^ Carl Ludwig Siegel : Lectures on Complex Function Theory . Volume 2. Wiley-Interscience, 1971, p. 163