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There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. 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 ...
The theta function ratios provide an efficient way of computing the Jacobi elliptic functions. There is an alternative method, based on the arithmetic-geometric mean and Landen's transformations : [ 6 ]
Theta functions are of great importance in mathematical physics because of their role in the inverse problem for periodic and quasi-periodic flows. The equations of motion are integrable in terms of Jacobi's elliptic functions in the well-known cases of the pendulum , the Euler top , the symmetric Lagrange top in a gravitational field , and the ...
There are a number of notational systems for the Jacobi theta functions.The notations given in the Wikipedia article define the original function (;) = = (+)which is equivalent to
In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. [1] [2] ...
In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as zn ( u , k ) {\displaystyle \operatorname {zn} (u,k)} [ 1 ]
If n = 1 and a and b are both 0 or 1/2, then the functions θ a,b (τ,z) are the four Jacobi theta functions, and the functions θ a,b (τ,0) are the classical Jacobi theta constants. The theta constant θ 1/2,1/2 (τ,0) is identically zero, but the other three can be nonzero.
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