Theta function

They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons.

Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian,[1] namely the Siegel upper half space.

In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.

One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has a positive imaginary part.

Accordingly, the theta function is 1-periodic in z: By completing the square, it is also τ-quasiperiodic in z, with Thus, in general, for any integers a and b.

It is in fact the most general entire function with 2 quasi-periods, in the following sense:[3] Theorem — If

The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript: The auxiliary (or half-period) functions are defined by This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ.

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.

Expanding terms out, the Jacobi triple product can also be written which we may also write as This form is valid in general but clearly is of particular interest when z is real.

as this integral identity: This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.

, then the following theta functions have interesting arithmetical and modular properties.

See Ramanujan's lost notebook and a relevant reference at Euler function.

values can be represented in a simplified way by using the hyperbolic lemniscatic sine: With the letter

Furthermore, those transformations are valid: These formulas can be used to compute the theta values of the cube of the nome: And the following formulas can be used to compute the theta values of the fifth power of the nome: The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations: The Rogers-Ramanujan continued fraction can be defined in terms of the Jacobi theta function in the following way: The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities: The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself.

The infinite sum[10][11] of the reciprocals of Fibonacci numbers with odd indices has this identity: By not using the theta function expression, following identity between two sums can be formulated: Also in this case

Infinite sum of the reciprocals of the Fibonacci number squares: Infinite sum of the reciprocals of the Pell numbers with odd indices: The next two series identities were proved by István Mező:[12] These relations hold for all 0 < q < 1.

The elliptic modulus is and the complementary elliptic modulus is These are two identical definitions of the complete elliptic integral of the second kind: The derivatives of the Theta Nullwert functions have these MacLaurin series: The derivatives of theta zero-value functions[14] are as follows: The two last mentioned formulas are valid for all real numbers of the real definition interval:

For the theta functions these integrals[15] are valid: The final results now shown are based on the general Cauchy sum formulas.

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.

[16] Taking z = x to be real and τ = it with t real and positive, we can write which solves the heat equation This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic delta function, or Dirac comb, in the sense of distributions General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 with the theta function.

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.

If F is a quadratic form in n variables, then the theta function associated with F is with the sum extending over the lattice of integers

For χ a primitive Dirichlet character modulo q and ν = ⁠1 − χ(−1)/2⁠ then is a weight ⁠1/2⁠ + ν modular form of level 4q2 and character which means[17] whenever Let be the set of symmetric square matrices whose imaginary part is positive definite.

One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking τ to be the period matrix with respect to a canonical basis for its first homology group.

The Riemann theta converges absolutely and uniformly on compact subsets of

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: In general, for all natural numbers

this formula of the Euler beta function is valid: In the following some Elliptic Integral Singular Values[18] are derived: The ensuing function has the following lemniscatically elliptic antiderivative: For the value

On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now: The regular partition sequence

can be written with the regular partition sequence P[24] and the strict partition sequence Q[25] can be generated like this: In the following table of sequences of numbers, this formula should be used as an example: Related to this property, the following combination of two series of sums can also be set up via the function ϑ01: Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3.