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1: Sidebar 21.SB1: Periodic Surface Waves
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. …
2: 25.13 Periodic Zeta Function
§25.13 Periodic Zeta Function
The notation F ( x , s ) is used for the polylogarithm Li s ( e 2 π i x ) with x real:
25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
F ( x , s ) is periodic in x with period 1, and equals ζ ( s ) when x is an integer. Also, …
3: 21.8 Abelian Functions
An Abelian function is a 2 g -fold periodic, meromorphic function of g complex variables. …For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
4: 24.17 Mathematical Applications
24.17.3 S n ( x ) = E ~ n ( x + 1 2 n + 1 2 ) E ~ n ( 1 2 n + 1 2 ) , n = 0 , 1 , ,
24.17.5 M n ( x ) = { B ~ n ( x ) - B n , n  even , B ~ n ( x + 1 2 ) , n  odd .
24.17.8 F ( x ) = B ~ n ( x ) - 2 - n B n
5: 25.1 Special Notation
k , m , n nonnegative integers.
B ~ n ( x ) periodic Bernoulli function B n ( x - x ) .
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
6: 24.2 Definitions and Generating Functions
§24.2(iii) Periodic Bernoulli and Euler Functions
B ~ n ( x ) = B n ( x ) ,
E ~ n ( x ) = E n ( x ) , 0 x < 1 ,
B ~ n ( x + 1 ) = B ~ n ( x ) ,
E ~ n ( x + 1 ) = - E ~ n ( x ) , x .
7: 29.19 Physical Applications
Simply-periodic Lamé functions ( ν 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. …
8: 29.10 Lamé Functions with Imaginary Periods
§29.10 Lamé Functions with Imaginary Periods
The first and the fourth functions have period 2 i K ; the second and the third have period 4 i K . …
9: 4.28 Definitions and Periodicity
§4.28 Definitions and Periodicity
Periodicity and Zeros
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . …
10: 21.3 Symmetry and Quasi-Periodicity
§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, …