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Sidebar 21.SB1: Periodic Surface Waves… ►Two-dimensional periodic waves in a shallow water wave tank. Taken from Joe Hammack, Daryl McCallister, Norman Scheffner and Harvey Segur, “Two-dimensional periodic waves in shallow water. …
§25.13 Periodic Zeta Function►The notation is used for the polylogarithm with real: ►
25.13.1… ► is periodic in with period 1, and equals when is an integer. Also, …
… ►An Abelian function is a -fold periodic, meromorphic function of complex variables. …For every Abelian function, there is a positive integer , such that the Abelian function can be expressed as a ratio of linear combinations of products with factors of Riemann theta functions with characteristics that share a common period lattice. …
►The main related functions are the Hurwitz zeta function , the dilogarithm , the polylogarithm (also known as Jonquière’s function ), Lerch’s transcendent , and the Dirichlet -functions .
periodic Bernoulli function .
§21.3 Symmetry and Quasi-Periodicity… ►This is the quasi-periodicity property of the Riemann theta function. … … ► …For Riemann theta functions with half-period characteristics, …
… ►Simply-periodic Lamé functions ( noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …