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1: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …

2: 22.10 Maclaurin Series

… ►

22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

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See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

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See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime }$

… ►The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime }$-plane in the case of (22.10.7)–(22.10.9); see §22.17.

3: 4.14 Definitions and Periodicity

… ►In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. … ►For $k\in \mathbb{Z}$ …

4: 22.17 Moduli Outside the Interval [0,1]

… ►In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in $k$. … ►In consequence, the formulas in this chapter remain valid when $k$ is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of $k$, irrespective of which values of $\sqrt{k}$ and $k^{\prime }=\sqrt{1-k^2}$ are chosen—as long as they are used consistently. …

5: 26.12 Plane Partitions

… ►An equivalent definition is that a plane partition is a finite subset of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$ with the property that if $(r,s,t)\in \pi$ and $(1,1,1)\le (h,j,k)\le (r,s,t)$, then $(h,j,k)$ must be an element of $\pi$. …It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. … ►

26.12.4 $$\prod _{(h,j,k)\in B(r,s,t)}\frac{h+j+k-1}{h+j+k-2}=\prod _{h=1}^r\prod _{j=1}^s\frac{h+j+t-1}{h+j-1}.$$

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Symbols:

$\in$: element of, $r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer, $j$: positive integer, $k$: positive integer and $B(r,s,t)$: Box

Permalink:

http://dlmf.nist.gov/26.12.E4

Encodings:

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See also:

Annotations for §26.12(i), §26.12 and Ch.26

►A plane partition is symmetric if $(h,j,k)\in \pi$ implies that $(j,h,k)\in \pi$. … ►A plane partition is cyclically symmetric if $(h,j,k)\in \pi$ implies $(j,k,h)\in \pi$. …

6: 24.17 Mathematical Applications

… ►

24.17.1 $$\sum _{j=a}^{n-1}{(-1)}^{j}f(j+h)=\frac{1}{2}\sum _{k=0}^{m-1}\frac{E_k\left(h\right)}{k!}\left({(-1)}^{n-1}{f}^{(k)}(n)+{(-1)}^a{f}^{(k)}(a)\right)+R_m(n),$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $m$: integer, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E1

Encodings:

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See also:

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

… ►Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. … ►

24.17.4 $$S_n(k)={(-1)}^k,$$ $k\in \mathbb{Z}$.

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Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.17.E4

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See also:

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

… ►

24.17.6 $$M_n(k)=0,$$ $k\in \mathbb{Z}$.

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Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer, $n$: integer and $M_n(x)$: function

Referenced by:

§24.17(ii)

Permalink:

http://dlmf.nist.gov/24.17.E6

Encodings:

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See also:

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

…

7: 21.6 Products

… ►that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. … ►

21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

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See also:

Annotations for §21.6(i), §21.6 and Ch.21

… ►

21.6.4 $$\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\sum }_{k=1}^h{T}_{jk}{𝐜}_k}{{\sum }_{k=1}^h{T}_{jk}{𝐝}_k}\right]\left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{-2\pi \mathrm{i}{\sum }_{j=1}^h{𝐛}_j\cdot {𝐜}_j}\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{𝐚}_j+{𝐜}_j}{{𝐛}_j+{𝐝}_j}\right]\left({𝐳}_j\right|𝛀),$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

…

8: 22.1 Special Notation

… ► ►►►►

$x,y$	real variables.
…
$k$	modulus. Except in §§22.3(iv), 22.17, and 22.19, $0\le k\le 1$.
$k^{\prime }$	complementary modulus, $k^2+k_{}^{\prime }{}_{}{}^2=1$. If $k\in [0,1]$, then $k^{\prime }\in [0,1]$.
…

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. …

9: 19.39 Software

… ►In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter. …References to research software that is available in other ways is listed separately. … ►Unless otherwise stated, the functions are $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. ►For research software see Bulirsch (1969b, function $\mathrm{cel}$), Herndon (1961a, b), Merner (1962), Morita (1978, complex modulus $k$), and Thacher Jr. (1963). … ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). …

10: 4.8 Identities

… ►

4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

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See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.11 $$\mathrm{Ln}\left(a^z\right)=z\mathrm{Ln}a+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer, $a$: real or complex constant and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►where the integer $k$ is chosen so that $\mathrm{\Re }\left(-\mathrm{i}z\ln a\right)+2k\pi \in [-\pi ,\pi ]$. …