Theta Functions - Theta Function (Paperback)

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, theta functions are special functions of several complex variables. They are important in several areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory, specifically string theory and D-branes. A theta function is graphed on a polar coordinate system. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this comes from a line bundle condition of descent. 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, where z can be any complex number and is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula where q=exp(i) and =exp(2iz). It is a Jacobi form. If is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity The function also behaves very regularly with respect to its quasi-period and satisfies the functional equation where a and b are integers. Theta function with different nome . The black dot in the right-hand picture indicates how is changing. Theta function with different nome . The black dot in the right-hand p... More: http: //booksllc.net/?id=449745