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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.
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 ...
For the above isosceles triangle with unit sides and angle , the area 1 / 2 × base × height is calculated in two orientations. When upright, the area is sin θ cos θ {\displaystyle \sin \theta \cos \theta } .
For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have < <. For negative values of θ we have, by the symmetry of the sine function
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [1] In the table below, the label "Undefined" represents a ratio :
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
Both the use of symbols and the naming order of tuple coordinates differ among the several sources and disciplines. This article will use the ISO convention [1] frequently encountered in physics, where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or (,,). (See graphic re the "physics convention".)
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