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1: 25.13 Periodic Zeta Function

§25.13 Periodic Zeta Function

►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

… ►Also, ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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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 $n(modk)$ and has the finite Fourier-series expansion …

3: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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${\tilde{B}}_n\left(x\right)$	periodic Bernoulli function $B_n\left(x-\lfloor x\rfloor \right)$.
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… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

4: 4.28 Definitions and Periodicity

§4.28 Definitions and Periodicity

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Periodicity and Zeros

►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

5: 4.14 Definitions and Periodicity

§4.14 Definitions and Periodicity

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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. …

7: 24.2 Definitions and Generating Functions

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§24.2(iii) Periodic Bernoulli and Euler Functions

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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 ${𝐦}_1+𝛀{𝐦}_2$ 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 …