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Jacobi Theta Functions (notational Variations)
There are a number of notational systems for the Theta function, Jacobi theta functions. The notations given in the Wikipedia article define the original function : \vartheta_(z; \tau) = \sum_^\infty \exp (\pi i n^2 \tau + 2 \pi i n z) which is equivalent to : \vartheta_(w, q) = \sum_^\infty q^ w^ where q=e^ and w=e^. However, a similar notation is defined somewhat differently in Whittaker and Watson, p. 487: : \vartheta_(x) = \sum_^\infty q^ \exp (2 \pi i n x/a) This notation is attributed to "Hermite, H.J.S. Smith and some other mathematicians". They also define : \vartheta_(x) = \sum_^\infty (-1)^n q^ \exp (\pi i (2 n + 1) x/a) This is a factor of ''i'' off from the definition of \vartheta_ as defined in the Wikipedia article. These definitions can be made at least proportional by ''x'' = ''za'', but other definitions cannot. Whittaker and Watson, Abramowitz and Stegun, and Gradshteyn and Ryzhik all follow Tannery and Molk, in which : \vartheta_1(z) = -i \sum_^\infty (-1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they 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 ), 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 quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions". Throughout this article, (e^)^ sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
