# of periodic functions

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##### 1: 25.13 Periodic Zeta Function
###### §25.13 Periodic Zeta Function
The notation $F\left(x,s\right)$ is used for the polylogarithm $\operatorname{Li}_{s}\left(e^{2\pi ix}\right)$ with $x$ real: Also,
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
##### 2: 27.10 Periodic Number-Theoretic Functions
###### §27.10 Periodic Number-Theoretic Functions
If $k$ is a fixed positive integer, then a number-theoretic function $f$ is periodic (mod $k$) if … Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form …where $g(m)$ is also periodic (mod $k$), and is given by … is a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion …
##### 3: 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 $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (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 $L\left(s,\chi\right)$.
##### 4: 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 $\pi i$. …
##### 6: 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. …
##### 8: 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. …
##### 9: 21.3 Symmetry and Quasi-Periodicity
This is the quasi-periodicity property of the Riemann theta function. …The set of points $\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function. … …For Riemann theta functions with half-period characteristics, …
##### 10: 28.30 Expansions in Series of Eigenfunctions
Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series …