# periods

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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 $F\left(x,s\right)$ is used for the polylogarithm $\mathrm{Li}_{s}\left(e^{2\pi ix}\right)$ with $x$ real: $F\left(x,s\right)$ is periodic in $x$ with period 1, and equals $\zeta\left(s\right)$ when $x$ is an integer. Also, …
##### 3: 21.8 Abelian Functions
An Abelian function is a $2g$-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
24.17.2 $R_{m}(n)=\frac{1}{2(m-1)!}\int_{a}^{n}f^{(m)}(x)\widetilde{E}_{m-1}\left(h-x% \right)\mathrm{d}x.$
24.17.3 $S_{n}(x)=\frac{\widetilde{E}_{n}\left(x+\tfrac{1}{2}n+\tfrac{1}{2}\right)}{% \widetilde{E}_{n}\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)},$ $n=0,1,\dots$,
24.17.5 $M_{n}(x)=\begin{cases}\widetilde{B}_{n}\left(x\right)-B_{n},&n\text{ even},\\ \widetilde{B}_{n}\left(x+\frac{1}{2}\right),&n\text{ odd}.\end{cases}$
##### 5: 25.1 Special Notation
 $k,m,n$ nonnegative integers. … periodic Bernoulli function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. …
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\mathrm{Li}_{2}\left(z\right)$, the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 6: 24.2 Definitions and Generating Functions
###### §24.2(iii) Periodic Bernoulli and Euler Functions
$\widetilde{B}_{n}\left(x\right)=B_{n}\left(x\right)$ ,
$\widetilde{E}_{n}\left(x\right)=E_{n}\left(x\right)$ , $0\leq x<1$,
$\widetilde{B}_{n}\left(x+1\right)=\widetilde{B}_{n}\left(x\right),$
$\widetilde{E}_{n}\left(x+1\right)=-\widetilde{E}_{n}\left(x\right)$ , $x\in\mathbb{R}$.
##### 7: 29.19 Physical Applications
Simply-periodic Lamé functions ($\nu$ 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 $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 9: 4.28 Definitions and Periodicity
###### Periodicity and Zeros
The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. …
##### 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, …