|
Jacobi Theta Functions (notational Variations)
There are a number of notational systems for the 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)^n q^ \exp ((2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
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. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. 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 do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
 |
Whittaker And Watson
''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1915. The first edition was Whittaker's alone, but later editions were co-authored with Watson. History Its first, second, third, and the fourth edition were published in 1902, 1915, 1920, and 1927, respectively. Since then, it has continuously been reprinted and is still in print today. A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021. The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final honou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |