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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.
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O. Sine function on unit circle (top) and its graph (bottom) In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle.
This formula applies to any algebraic equation of any degree without need for a Tschirnhaus transformation or any other manipulation to bring the equation into a specific normal form, such as the Bring–Jerrard form for the quintic. However, application of this formula in practice is difficult because the relevant hyperelliptic integrals and ...
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)}
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 ...
In mathematics, an elliptic hypergeometric series is a series Σc n such that the ratio c n /c n−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n.