complete integrals

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1: 29.10 Lamé Functions with Imaginary Periods

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29.10.2 $$z^{\prime }=\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),$$

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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, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

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${\mathrm{𝐸𝑐}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

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${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k_{}^{\prime }{}_{}{}^2),$

… ►The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

2: 29.13 Graphics

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Figure 29.13.5: ${\mathrm{𝑢𝐸}}_4^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.13.6: ${\mathrm{𝑢𝐸}}_4^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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Figure 29.13.21: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.1)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=1.61244\mathrm{\dots }$, $K^{\prime }=2.57809\mathrm{\dots }$. Magnify 3D Help

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Figure 29.13.22: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.5)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=K^{\prime }=1.85407\mathrm{\dots }$. Magnify 3D Help

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Figure 29.13.23: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.9)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. $K=2.57809\mathrm{\dots }$, $K^{\prime }=1.61244\mathrm{\dots }$. Magnify 3D Help

3: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …

4: 22.4 Periods, Poles, and Zeros

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Table 22.4.1: Periods and poles of Jacobian elliptic functions.

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Periods	$z$-Poles
	$\mathrm{i}{K}^{\prime }$	$K+\mathrm{i}{K}^{\prime }$	$K$	$0$
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Table 22.4.2: Periods and zeros of Jacobian elliptic functions.

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Periods	$z$-Zeros
	$0$	$K$	$K+\mathrm{i}{K}^{\prime }$	$\mathrm{i}{K}^{\prime }$
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►Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2\mathrm{i}{K}^{\prime }$, $2\mathrm{i}{K}^{\prime }$. … ►Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\text{s}=0$, $\text{c}=K$, $\text{d}=K+\mathrm{i}{K}^{\prime }$, $\text{n}=\mathrm{i}{K}^{\prime }$. … ►This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. …

5: 22.3 Graphics

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Figure 22.3.16: $\mathrm{sn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.17: $\mathrm{cn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.18: $\mathrm{dn}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.19: $\mathrm{cd}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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Figure 22.3.20: $\mathrm{dc}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. $K=3.3566\mathrm{\dots }$, $K^{\prime }=1.5786\mathrm{\dots }$. Magnify 3D Help

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6: 29.14 Orthogonality

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29.14.2 $$⟨g,h⟩={\int }_0^K{\int }_0^{K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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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, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.3 $$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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29.14.11 $$⟨g,h⟩={\int }_0^{4K}{\int }_0^{2K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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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, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Permalink:

http://dlmf.nist.gov/29.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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7: 29.17 Other Solutions

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29.17.1 $$F(z)=E(z){\int }_{\mathrm{i}{K}^{\prime }}^z\frac{du}{{(E(u))}^2}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/29.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.17(i), §29.17 and Ch.29

… ►They are algebraic functions of $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$, and have primitive period $8K$. … ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

8: 22.11 Fourier and Hyperbolic Series

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22.11.7 $$\mathrm{ns}(z,k)-\frac{\pi }{2K}\csc \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.10

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E7

Encodings:

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See also:

Annotations for §22.11 and Ch.22

… ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , ►

22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.14 $$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$

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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, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. …

9: 29.4 Graphics

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Figure 29.4.13: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.5)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.85407\mathrm{\dots }$. Magnify

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Figure 29.4.14: ${\mathrm{𝐸𝑠}}_{1.5}^m(x,0.5)$ for $-2K\le x\le 2K$, $m=1,2,3$. $K=1.85407\mathrm{\dots }$. Magnify

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Figure 29.4.15: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.4.16: ${\mathrm{𝐸𝑠}}_{1.5}^m(x,0.1)$ for $-2K\le x\le 2K$, $m=1,2,3$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.4.17: ${\mathrm{𝐸𝑐}}_{1.5}^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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10: 19.3 Graphics

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Figure 19.3.1: $K\left(k\right)$ and $E\left(k\right)$ as functions of $k^2$ for $-2\le k^2\le 1$. Graphs of $K^{\prime }\left(k\right)$ and $E^{\prime }\left(k\right)$ are the mirror images in the vertical line $k^2=\frac{1}{2}$. Magnify

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. …As ${\alpha }^2\to 1+$ it has the limit $K\left(k\right)-(E\left(k\right)/k_{}^{\prime }{}_{}{}^2)$. If ${\alpha }^2=0$, then it reduces to $K\left(k\right)$. … Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2>1$) it has the (negative) value $K\left(k^{\prime }\right)-E\left(k^{\prime }\right)$, with limit 0 as $k^2\to 1+$. Magnify 3D Help