# Theta Function

(redirected from Riemann theta function)
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Related to Riemann theta function: Riemann hypothesis, Jacobi theta functions

## Theta Function

Theta functions are entire functions whose quotients are elliptic functions.

The four principal theta functions are defined by the following rapidly converging series:

θ1(z) = 2q1/4 sin z – 2q9/4 sin 3z + 2q25/4 sin 5z – . . .

θ2(z) = 2q1/4 cos z – 2q9/4 cos 3z + 2q25/4 cos 5z + . . .

θ3(z) = 1 + 2q cos 2z + 2q4 cos 4z + 2q9 cos 6z + . . .

θ4(z) = 1 – 2q cos 2z + 2q4 cos 4z – 2q9 cos 6z + . . .

where ǀqǀ < 1.

When π is added to the argument z, the functions are multiplied by –1, –1, 1, and 1, respectively. When πτ is added to z, the relation between τ and q being given by the equation q = eπiτ the functions are multiplied by the factors – N, N, N, and –N, respectively, where N = q–1e–2ik. It follows that, for example, the quotient θ1(Z)/θ4(Z) is a meromorphic function that does not change when 2π or irr is added to the argument—that is, this quotient is an elliptic function with periods 2π and πτ.

For the representation of automorphic functions, H. Poincaré constructed theta functions that are a generalization of the theta functions described above, which were introduced by K. Jacobi (Jacobi’s notation was somewhat different).

### REFERENCE

Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd edition, part 2. Moscow, 1963. (Translated from English.)
References in periodicals archive ?
where m [member of] Z, complex parameter s, [epsilon] [member of] C, and complex phase variables [xi] [member of] C, [tau] > 0 which is called the period matrix of the Riemann theta function.
Next, we turn to see the periodicity of the solution (23); the function f is chosen to be a Riemann theta function; namely,
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In this paper, based on the Hirotas bilinear method, combining the theory of a general Riemann theta function, we have derived a method of constructing double periodic wave solutions for (2 + 1)-dimensional nonlinear partial differential equations.
Assuming that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are two Riemann theta functions with [xi] = [alpha]x + [beta]y + wt + [sigma], then Hirota bilinear operators [D.sub.x], [D.sub.y], and [D.sub.t] exhibit the following perfect properties when they act on a pair of theta functions:
And then, we construct the exact periodic wave solution of BLMP equation with the aid of the Riemann theta function, Hirota direct method, and the special property of the [D.sub.p]-operators when acting on exponential functions.
Furthermore, together with Riemann theta function and Hirota method, we successfully get the exact periodic wave solution and figure of BLMP equation when p = 5.

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