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

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

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

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

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

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Figure 29.13.21: $|{\mathit{uE}}_4^1(x+\mathrm{i}y,0.1)|$ 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: $|{\mathit{uE}}_4^1(x+\mathrm{i}y,0.5)|$ 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: $|{\mathit{uE}}_4^1(x+\mathrm{i}y,0.9)|$ 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: 10.35 Generating Function and Associated Series

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10.35.2 $${\mathrm{e}}^{z\cos \theta }=I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}{I}_k\left(z\right)\cos \left(k\theta \right),$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.34

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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10.35.4 $$1=I_0\left(z\right)-2I_2\left(z\right)+2I_4\left(z\right)-2I_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.36

Permalink:

http://dlmf.nist.gov/10.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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10.35.5 $${\mathrm{e}}^{\pm z}=I_0\left(z\right)\pm 2I_1\left(z\right)+2I_2\left(z\right)\pm 2I_3\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.37,9.6.38

Permalink:

http://dlmf.nist.gov/10.35.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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$$\cosh z=I_0\left(z\right)+2I_2\left(z\right)+2I_4\left(z\right)+2I_6\left(z\right)+\mathrm{\dots },$$

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$$\sinh z=2I_1\left(z\right)+2I_3\left(z\right)+2I_5\left(z\right)+\mathrm{\dots }.$$

4: 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. …

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: 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 $\mathrm{p}\mathrm{q}(z,k)$. …

7: 10.28 Wronskians and Cross-Products

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10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

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10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

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Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

8: 14.10 Recurrence Relations and Derivatives

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14.10.2 $${\left(1-x^2\right)}^{1/2}{\mathsf{P}}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1){\mathsf{P}}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{\mathsf{P}}_{\nu }^{\mu }\left(x\right)=0,$$

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Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.3 $$(\nu -\mu +2){\mathsf{P}}_{\nu +2}^{\mu }\left(x\right)-(2\nu +3)x{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1){\mathsf{P}}_{\nu }^{\mu }\left(x\right)=0,$$

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Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.3 (modified)

Referenced by:

§14.10, §14.14, §14.21(iii)

Permalink:

http://dlmf.nist.gov/14.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.5 $$\left(1-x^2\right)\frac{d{\mathsf{P}}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){\mathsf{P}}_{\nu -1}^{\mu }\left(x\right)-\nu x{\mathsf{P}}_{\nu }^{\mu }\left(x\right).$$

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Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.4 (modified)

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

► ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ also satisfies (14.10.1)–(14.10.5). …In addition, $P_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$ satisfy (14.10.3)–(14.10.5).

9: 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, $\mathrm{i}$: 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 $2K+\mathrm{i}{K}^{\prime }$. …

10: 14.33 Tables

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Abramowitz and Stegun (1964, Chapter 8) tabulates ${\mathsf{P}}^n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; $\mathsf{P}^n{}^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${\mathsf{Q}}^n\left(x\right)$ and $\mathsf{Q}^n{}^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P^n\left(x\right)$ and $P^n{}^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q^n\left(x\right)$ and $Q^n{}^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

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Zhang and Jin (1996, Chapter 4) tabulates ${\mathsf{P}}^n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${\mathsf{P}}^n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${\mathsf{Q}}^n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${\mathsf{Q}}^n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${\mathsf{P}}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${\mathsf{Q}}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${\mathsf{P}}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P^n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q^n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${\mathsf{P}}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

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Belousov (1962) tabulates ${\mathsf{P}}_n^m\left(\cos \theta \right)$ (normalized) for $m=0(1)36$, $n-m=0(1)56$, $\theta =0({2.5}^{\circ }){90}^{\circ }$, 6D.

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Žurina and Karmazina (1964, 1965) tabulate the conical functions ${\mathsf{P}}^{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P^{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ for $\tau =0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when .

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Žurina and Karmazina (1963) tabulates the conical functions ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau }^1\left(x\right)$ for $\tau =0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when .

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