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Theta function

In mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the behavior of discrete multi-dimensional periodic systems, such as crystal lattices or points on a torus. Because they are smooth, they allow the study and manipulation of discrete combinatorial…

Generalizations, Jacobi theta function & Basic example

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Overview

Basic example

Jacobi theta function

Auxiliary functions

Jacobi identities

Theta functions in terms of the nome

Product representations

Explicit values

Nome power theorems

Some series identities

Relation to the Riemann zeta function

Relation to the Weierstrass elliptic function

Relation to the q-gamma function

Relations to Dedekind eta function

Elliptic modulus

Derivatives of theta functions

A solution to the heat equation

Relation to the Heisenberg group

Generalizations

Derivation of the theta values

Partition sequences and Pochhammer products

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Theta function

Nodes139
Edges138
Triples104
Avg. degree1.99
Density0.014388
Components1

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Theta function

Top relations

related to Further reading · 40
Theta function → Acad, Ackerman, Baltimore MD, BF01420339, Bruno, Charles Hermite, Comptes, Developments, Die Grundlehren, Eisenstein, Elliptic, Farkas, February, Harry Rauch, Hershel, In Alladi, ISBN, IX, Krishnaswami, March
related to External links · 9
Theta function → Creative Commons Attribution/Share-Alike License, Elliptic, Integral, Jacobi, Matlab, Moiseev Igor, Octave, PlanetMath, This
related to Equianharmonic values · 5
Theta function → Bigl, Bigr, Bruce Berndt, Gamma, The
related to Jacobi theta function · 5
Theta function → Carl Gustav Jacob Jacobi, It, Jacobi, One Jacobi, There
related to Application to elliptic functions · 4
Theta function → Abstractly, One, Theta, With
related to Further values · 4
Theta function → Bigl, Bigr, Gamma, Many
is a · 3
Theta function → fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions, modular form of weight, special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions
related to Identity of the Euler beta function · 3
Theta function → Euler, In, This
related to Jacobi identities · 3
Theta function → Equations, For, Jacobi's
related to Relation to the Heisenberg group · 3
Theta function → Heisenberg, The Jacobi, This

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Important terminology

theta displaystyle function functions pi tau infty sum left frac vartheta right jacobi exp 10 -1 nome begin end 01

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Theta functionis aspecial function that describes the evolution of temperature on a segment domain subject to certain boundary conditions0.90text
Theta functionis afundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions0.90text
Theta functionis amodular form of weight0.90text
Theta functionrelated to A solution to the heat equationThe Jacobi0.60section
Theta functionrelated to A solution to the heat equationTaking0.60section
Theta functionrelated to Application to elliptic functionsTheta0.60section
Theta functionrelated to Application to elliptic functionsWith0.60section
Theta functionrelated to Application to elliptic functionsAbstractly0.60section
Theta functionrelated to Application to elliptic functionsOne0.60section
Theta functionrelated to Auxiliary functionsThe Jacobi0.60section
Theta functionrelated to Basic exampleOne0.60section
Theta functionrelated to Direct power theoremsFor0.60section

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