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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 ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ 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

… ► $F(x,s)$ 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

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24.17.2 $$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$

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Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

… ►

24.17.3 $$S_n(x)=\frac{{\tilde{E}}_n\left(x+\frac{1}{2}n+\frac{1}{2}\right)}{{\tilde{E}}_n\left(\frac{1}{2}n+\frac{1}{2}\right)},$$ $n=0,1,\mathrm{\dots }$,

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Symbols:

${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

… ►

24.17.5

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Symbols:

$B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer, $x$: real or complex and $M_n(x)$: function

Permalink:

http://dlmf.nist.gov/24.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

… ►

24.17.8 $$F(x)={\tilde{B}}_n\left(x\right)-2^{-n}{B}_n$$

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Symbols:

$B_n$: Bernoulli numbers, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

…

5: 25.1 Special Notation

… ► ►►►

$k,m,n$	nonnegative integers.
…
${\tilde{B}}_n\left(x\right)$	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 $L(s,\chi )$.

6: 24.2 Definitions and Generating Functions

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§24.2(iii) Periodic Bernoulli and Euler Functions

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${\tilde{B}}_n\left(x\right)=B_n\left(x\right)$ ,

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${\tilde{E}}_n\left(x\right)=E_n\left(x\right)$ , ,

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${\tilde{B}}_n\left(x+1\right)={\tilde{B}}_n\left(x\right),$

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${\tilde{E}}_n\left(x+1\right)=-{\tilde{E}}_n\left(x\right)$ , $x\in ℝ$.

…

7: 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, …

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. … ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …

9: 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 $4\mathrm{i}{K}^{\prime }$. …

10: 4.28 Definitions and Periodicity

§4.28 Definitions and Periodicity

… ►

Periodicity and Zeros

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