Theta function

This article deals with the Jacobian theta functions. For more meanings of the term, see theta function (disambiguation) .

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 . ${\displaystyle \Theta }$

Theta functions appear, for example, when solving the heat conduction equation .

definition

Classic theta function

The classic Jacobean theta function is defined by

${\displaystyle \vartheta (z,\tau ):=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau +2\pi inz}}$

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 . ${\displaystyle \mathbb {C} \times \mathbb {H} }$ ${\displaystyle \mathbb {H} =\{z\in \mathbb {C} \mid \Im (z)>0\}}$ ${\displaystyle \tau \in \mathbb {H} }$ ${\displaystyle \vartheta (\cdot ,\tau )}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle \vartheta (z,\cdot )}$ ${\displaystyle \mathbb {H} }$

More theta functions

In addition to the classic theta function, there are three other theta functions in the literature, namely:

${\displaystyle \vartheta _{0}(z,\tau ):=\vartheta _{0,1}(z,\tau ):=\vartheta \left(z+{\frac {1}{2}},\tau \right)=\sum _{n=-\infty }^{\infty }(-1)^{n}\cdot e^{\pi in^{2}\tau +2\pi inz}}$

${\displaystyle \vartheta _{2}(z,\tau ):=\vartheta _{1,0}(z,\tau ):=e^{\pi i{\frac {\tau }{4}}+\pi iz}\cdot \vartheta \left(z+{\frac {\tau }{2}},\tau \right)=\sum _{n=-\infty }^{\infty }e^{\pi i\left(n+{\frac {1}{2}}\right)^{2}\tau +2\pi i\left(n+{\frac {1}{2}}\right)z}}$

${\displaystyle \vartheta _{1}(z,\tau ):=\vartheta _{1,1}(z,\tau ):=e^{\pi i{\frac {\tau }{4}}+\pi i\left(z+{\frac {1}{2}}\right)}\cdot \vartheta \left(z+{\frac {\tau +1}{2}},\tau \right)=i\cdot \sum _{n=-\infty }^{\infty }(-1)^{n}\cdot e^{\pi i\left(n+{\frac {1}{2}}\right)^{2}\tau +2\pi i\left(n+{\frac {1}{2}}\right)z}}$

The Jacobian theta function is referred to as or in this notation . ${\displaystyle \vartheta _{3}(z,\tau )}$ ${\displaystyle \vartheta _{0,0}(z,\tau )}$

One defines something more general

${\displaystyle \Theta _{a,b}(z,\tau ):=\sum _{n=-\infty }^{\infty }e^{\pi i\left(n+{\frac {a}{2}}\right)^{2}\tau +2\pi i(n+{\frac {a}{2}})z+\pi inb}}$

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 ${\displaystyle z=0}$

${\displaystyle \vartheta (\tau ):=\vartheta (0,\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }=1+2\sum _{n=1}^{\infty }e^{\pi in^{2}\tau }}$

properties

zeropoint

For solid , the theta function has simple zeros at the points ${\displaystyle \tau \in \mathbb {H} }$

${\displaystyle z=k+m\tau +{\frac {\tau +1}{2}},k,m\in \mathbb {Z} }$.

Transformation formula

The theta function is periodic in both variables, it is

${\displaystyle \vartheta (z+1,\tau )=\vartheta (z,\tau +2)=\vartheta (z,\tau )}$

In addition, the important transformation formula applies

${\displaystyle \vartheta \left(z,{\frac {-1}{\tau }}\right)=e^{\pi iz^{2}\tau }{\sqrt {\frac {\tau }{i}}}\vartheta (z\tau ,\tau )}$

This is especially reduced for the theta zero value

${\displaystyle \vartheta \left({\frac {-1}{\tau }}\right)={\sqrt {\frac {\tau }{i}}}\vartheta (\tau )}$

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:

${\displaystyle \vartheta (z,\tau )=\prod _{n=1}^{\infty }(1-e^{2\pi in\tau })(1+e^{\pi i[(2n-1)\tau +2z]})(1+e^{\pi i[(2n-1)\tau -2z]})}$

This is especially reduced for the theta zero value

${\displaystyle \vartheta (\tau )=\prod _{n=1}^{\infty }(1-e^{2\pi in\tau })(1+e^{\pi i(2n-1)\tau })^{2}}$

From this representation it follows in particular that there are no zeros in the upper half-plane . ${\displaystyle \vartheta (\tau )}$ ${\displaystyle \mathbb {H} }$

Integral representation

The theta function has an integral representation:

${\displaystyle \vartheta (z,\tau )=i\int _{i-\infty }^{i+\infty }{e^{i\pi \tau u^{2}}\cos(2\pi uz+\pi u) \over \sin(\pi u)}{\text{d}}u}$

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 ${\displaystyle x}$ ${\displaystyle t>0}$

${\displaystyle {\frac {\partial }{\partial t}}\vartheta (x,it)={\frac {1}{4\pi }}{\frac {\partial ^{2}}{\partial x^{2}}}\vartheta (x,it).}$

how to by inserting

 ${\displaystyle \vartheta (x,it)=\sum _{n=-\infty }^{\infty }e^{-n^{2}\cdot \pi \cdot t+2\pi inx}=1+2\sum _{n=1}^{\infty }e^{-n^{2}\cdot \pi \cdot t}\cos {(2\pi nx)}}$

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:

${\displaystyle \vartheta _{3}(\tau )^{4}=\vartheta _{0}(\tau )^{4}+\vartheta _{2}(\tau )^{4}}$

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:

${\displaystyle \Gamma \left({\frac {s}{2}}\right)\pi ^{-{\frac {s}{2}}}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }(\vartheta (0,it)-1)\,t^{\frac {s}{2}}{\frac {{\text{d}}t}{t}}}$

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:

${\displaystyle \vartheta (0,\tau )={\frac {\eta ^{2}({\frac {\tau +1}{2}})}{\eta (\tau +1)}}}$

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 ${\displaystyle f(\tau ):=\vartheta ^{8}(\tau )}$

${\displaystyle f(\tau +2)=f(\tau )\quad {\text{und}}\quad f\left({\frac {-1}{\tau }}\right)=\tau ^{4}f(\tau )}$

The function is therefore a modular form of weight 4 to the subgroup of the module group generated by the two transformations and . ${\displaystyle f(\tau )}$ ${\displaystyle \tau \mapsto \tau +2}$ ${\displaystyle \tau \mapsto {\frac {-1}{\tau }}}$ ${\displaystyle \Gamma _{\vartheta }}$ ${\displaystyle \Gamma }$

Quotients of theta functions

The theta function can be used to define elliptic functions . If one sets for solid : ${\displaystyle \tau \in \mathbb {H} }$

${\displaystyle f(z)={\frac {\vartheta ^{2}(z+{\frac {1}{2}},\tau )}{\vartheta ^{2}(z,\tau )}}}$,

so is an elliptic function to the lattice . ${\displaystyle f(z)}$ ${\displaystyle \mathbb {Z} +\mathbb {Z} \tau }$

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: ${\displaystyle f(z)}$ ${\displaystyle f(z+1)=f(z)}$ ${\displaystyle f(z+\tau )={\text{e}}^{-az-b}f(z)}$ ${\displaystyle \tau \in \mathbb {H} }$ ${\displaystyle \mathbb {Z} +\mathbb {Z} \tau }$

${\displaystyle \wp (z)=-{\frac {{\text{d}}^{2}}{{\text{d}}z^{2}}}\log \vartheta _{1}(z,\tau )+c}$

with a matching constant . ${\displaystyle c}$

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:

${\displaystyle \prod _{n=1}^{\infty }(1-q^{n})^{3}=\sum _{m=0}^{\infty }(-1)^{m}(2m+1)q^{(m^{2}+m)/2}}$

See also

• Ramanujan theta function

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

• Theta Functions in NIST Digital Library of Mathematical Functions
• Eric W. Weisstein : Jacobi Theta Functions . In: MathWorld (English).

Individual evidence

1. ^ 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
2. ^ Carl Ludwig Siegel : Lectures on Complex Function Theory . Volume 2. Wiley-Interscience, 1971, p. 163