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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 Neville theta functions are related to the Jacobi elliptic functions. If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications.
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa ...
Spaces within a formula must be directly managed (for example by including explicit hair or thin spaces). Variable names must be italicized explicitly, and superscripts and subscripts must use an explicit tag or template. Except for short formulas, the source of a formula typically has more markup overhead and can be difficult to read.
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 ] It is given by θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 − q n + 1 / z ) {\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)}
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
George Andrews [14] showed that several of Ramanujan's fifth order mock theta functions are equal to quotients Θ(𝜏) / θ(𝜏) where θ(𝜏) is a modular form of weight 1 / 2 and Θ(𝜏) is a theta function of an indefinite binary quadratic form, and Dean Hickerson [15] proved similar results for seventh order mock theta ...