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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^)^ should b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 1902. 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 hono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press, Inc
An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy, founded approximately 385 BC at Akademia, a sanctuary of Athena, the goddess of wisdom and skill, north of Athens, Greece. Etymology The word comes from the ''Academy'' in ancient Greece, which derives from the Athenian hero, '' Akademos''. Outside the city walls of Athens, the gymnasium was made famous by Plato as a center of learning. The sacred space, dedicated to the goddess of wisdom, Athena, had formerly been an olive grove, hence the expression "the groves of Academe". In these gardens, the philosopher Plato conversed with followers. Plato developed his sessions into a method of teaching philosophy and in 387 BC, established what is known today as the Old Academy. By extension, ''academia'' has come to mean the accumulatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Course In Modern Analysis
''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 1902. 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 hono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Functions
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^)^ should be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
