periodicity

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31: 2.10 Sums and Sequences

… ►As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$. … ►

2.10.1 $$\sum _{j=a}^{n}f(j)={\int }_a^{n}f(x)dx+\frac{1}{2}f(a)+\frac{1}{2}f(n)+\sum _{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^{(2s-1)}(n)-f^{(2s-1)}(a)\right)+{\int }_a^n\frac{B_{2m}-{\tilde{B}}_{2m}\left(x\right)}{(2m)!}{f}^{(2m)}(x)dx.$$

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Symbols:

$B_n$: Bernoulli numbers, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $a$: integer, $m$: integer, $n$: integer and $f(x)$: integrable function

Referenced by:

§2.10(i), §2.10(i)

Permalink:

http://dlmf.nist.gov/2.10.E1

Encodings:

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See also:

Annotations for §2.10(i), §2.10 and Ch.2

… ►

2.10.5 $$R_m(n)={\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2m}\left(x\right)-B_{2m}}{2m(2m-1)x^{2m-1}}dx.$$

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Symbols:

$B_n$: Bernoulli numbers, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $m$: integer, $n$: integer and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/2.10.E5

Encodings:

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See also:

Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

►From §24.12(i), (24.2.2), and (24.4.27), ${\tilde{B}}_{2m}\left(x\right)-B_{2m}$ is of constant sign ${(-1)}^m$. …

32: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

Table 29.3.1: Eigenvalues of Lamé’s equation.

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eigenvalue $h$	parity	period
…

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… ►They are called Lamé functions with real periods and of order $\nu$ , or more simply, Lamé functions. … ►

Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
…

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…

33: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … ►If $f(x)$ is of period $2\pi$, and $f^{(m)}(x)$ is piecewise continuous, then … ►If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$. … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

34: 31.2 Differential Equations

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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35: 4.16 Elementary Properties

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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
…

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…

36: 27.8 Dirichlet Characters

… ►If $k$ $(>1)$ is a given integer, then a function $\chi \left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. …

37: About Color Map

… ►For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. …

38: 21.2 Definitions

… ►

21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

►Characteristics whose elements are either $0$ or $\frac{1}{2}$ are called half-period characteristics. For given $𝛀$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics. …

39: 22.2 Definitions

… ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …

40: 28.12 Definitions and Basic Properties

… ►

28.12.6 $${\mathrm{me}}_{\nu }(z+\pi ,q)={\mathrm{e}}^{\pi \mathrm{i}\nu }{\mathrm{me}}_{\nu }(z,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.4 (in different form)

Permalink:

http://dlmf.nist.gov/28.12.E6

Encodings:

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See also:

Annotations for §28.12(ii), §28.12 and Ch.28

… ►When $\nu =s/m$ is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period $2m\pi$. …