|
In mathematics, theta functions are special functions of several complex variables. They are important in many 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. 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. ==Jacobi theta function== 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 confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula : where ''q'' = exp(π''i''τ) and η = exp(2π''iz''). It is a Jacobi form. If τ 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 τ and satisfies the functional equation : where ''a'' and ''b'' are integers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「theta function」の詳細全文を読む スポンサード リンク
|