complete integrals

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21: 29.2 Differential Equations

… ►This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K}^{\prime }$, where $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

29.2.8 $$\eta ={(e_1-e_3)}^{-1/2}(z-\mathrm{i}{K}^{\prime }),$$

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Defines:

$\eta$ (locally)

Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable, $k$: real parameter and $e_j$: constants

Permalink:

http://dlmf.nist.gov/29.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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22: 19.9 Inequalities

… ►

§19.9(i) Complete Integrals

… ►

19.9.5

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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, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

… ►

19.9.8

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

►Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►

19.9.9 $$L(a,b)=4aE\left(k\right),$$ $k^2=1-(b^2/a^2)$, $a>b$.

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $L(a,b)$: perimeter

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

…

23: 29.18 Mathematical Applications

… ►

$\beta =K+\mathrm{i}{\beta }^{\prime },$

►

$0\le {\beta }^{\prime }\le 2K^{\prime },$

►

$0\le \gamma \le 4K,$

… ►

$\alpha =K+\mathrm{i}{K}^{\prime }-{\alpha }^{\prime },$ ,

►

$\beta =K+\mathrm{i}{\beta }^{\prime },$ $0\le {\beta }^{\prime }\le 2K^{\prime },0\le \gamma \le 4K$,

…

24: 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
…

►

…

25: 29.8 Integral Equations

… ►Let $w(z)$ be any solution of (29.2.1) of period $4K$, $w_2(z)$ be a linearly independent solution, and $𝒲\left\{w,w_2\right\}$ denote their Wronskian. … ►

29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

$$w(z+2K)=\sigma w(z),$$

►

$$w_2(z+2K)=\tau w(z)+\sigma w_2(z).$$

… ►

29.8.5 $${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z_1,k^2)\frac{w_2(K)-w_2(-K)}{{d{w}_2(z)/dz|}_{z=0}}={\int }_{-K}^K{𝖯}_{\nu }\left(y\right){\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)dz,$$

ⓘ

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

26: 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 $[1,\mathrm{\infty })$. …

27: 19.5 Maclaurin and Related Expansions

… ►

19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

ⓘ

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, $q$: nome and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). ►

19.5.8 $$K\left(k\right)=\frac{\pi }{2}{\left(1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\right)}^2,$$ ,

ⓘ

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, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

►

19.5.9 $$E\left(k\right)=K\left(k\right)+\frac{2{\pi }^2}{K\left(k\right)}\frac{{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{n}^2{q}^{n^2}}{1+2{\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^n{q}^{n^2}},$$ .

ⓘ

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, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product …

28: 19.2 Definitions

… ►

$D\left(k\right)=D(\pi /2,k)=(K\left(k\right)-E\left(k\right))/k^2,$

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

19.2.8_1 $$K^{\prime }\left(k\right)={\int }_0^1\frac{dt}{\sqrt{1-t^2}\sqrt{1-(1-k^2)t^2}},$$

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Defines:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

►

19.2.8_2 $$E^{\prime }\left(k\right)={\int }_0^1\frac{\sqrt{1-(1-k^2)t^2}}{\sqrt{1-t^2}}dt,$$

ⓘ

Defines:

$E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral and $k$: real or complex modulus

Referenced by:

§19.2(ii)

Permalink:

http://dlmf.nist.gov/19.2.E8_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.10):

This equation was added.

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

… ►

$K\left(k\right)=\mathrm{cel}(k_c,1,1,1),$

…

29: 22.14 Integrals

… ►Thirdly, with , … ►Lastly, with , … ►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

►

22.14.17 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{cn}(t,k)\right)dt=-\frac{\pi }{4}{K}^{\prime }\left(k\right)+\frac{1}{2}K\left(k\right)\ln \left(k^{\prime }/k\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: 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, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

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

Permalink:

http://dlmf.nist.gov/22.14.E17

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

…

30: 23.7 Quarter Periods

… ►

23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►

23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►

23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

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