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31: 22.14 Integrals

… ►Thirdly, with , … ►Lastly, with , … ►The indefinite integral of a 4th power can be expressed as a complete elliptic integral, a polynomial in Jacobian functions, and the integration variable. … ►In (22.14.13)–(22.14.15), . … ►

22.14.16 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{sn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)-\frac{1}{2}K\left(k\right)\ln k,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $k$: modulus

Referenced by:

Erratum (V1.0.28) for Equations (22.14.16), (22.14.17)

Permalink:

http://dlmf.nist.gov/22.14.E16

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

Originally, a factor of $\pi$ was missing from the term containing the $\frac{1}{4}{K}^{\prime }\left(k\right)$.

Suggested 2020-08-06 by Fred Hucht

See also:

Annotations for §22.14(iv), §22.14 and Ch.22

…

32: 22.16 Related Functions

… ►where the inverse sine has its principal value when $-K\le x\le K$ and is defined by continuity elsewhere. … ►In Equations (22.16.21)–(22.16.23), … ►In Equations (22.16.24)–(22.16.26), . … ►For $E\left(k\right)$ see §19.2(ii). … ►For $E\left(k\right)$ see §19.2(ii). …

33: 29.12 Definitions

… ►The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$. … ►

Table 29.12.1: Lamé polynomials.

► ►►►►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
$2n$	$a_{\nu }^{2m}\left(k^2\right)$	${\mathrm{𝑢𝐸}}_{\nu }^m(z,k^2)$	$P({\mathrm{sn}}^2)$	$2K$	$2\mathrm{i}{K}^{\prime }$	even	even	even
$2n+1$	$a_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑠𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{sn}P({\mathrm{sn}}^2)$	$4K$	$2\mathrm{i}{K}^{\prime }$	odd	even	even
$2n+1$	$b_{\nu }^{2m+1}\left(k^2\right)$	${\mathrm{𝑐𝐸}}_{\nu }^m(z,k^2)$	$\mathrm{cn}P({\mathrm{sn}}^2)$	$4K$	$4\mathrm{i}{K}^{\prime }$	even	odd	even
…

►

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34: 19.39 Software

… ►A more complete list of available software for computing these functions is found in the Software Index. … ►

§19.39(ii) Legendre’s and Bulirsch’s Complete Integrals

►Unless otherwise stated, the functions are $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. … ►For other software, sometimes with $\mathrm{\Pi }({\alpha }^2,k)$ and complex variables, see the Software Index. …

35: 22.18 Mathematical Applications

… ►where $k=\sqrt{1-(b^2/a^2)}$ is the eccentricity, and $0\le u\le 4K\left(k\right)$. … ►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. …

36: 18.42 Software

… ►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …

37: 22.2 Definitions

… ►

22.2.1 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right),$$

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Defines:

$q$: nome

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function and $k$: modulus

Referenced by:

§20.9(i), §22.16(i)

Permalink:

http://dlmf.nist.gov/22.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►

$K\left(k\right)={\displaystyle \frac{\pi }{2}}{\theta }_3^2(0,q),$

… ►

22.2.3 $$\zeta =\frac{\pi z}{2K\left(k\right)},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►

22.2.12 $$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: modulus and $\tau$: ratio

Referenced by:

§22.2

Permalink:

http://dlmf.nist.gov/22.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

…

38: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

39: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

40: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. …