DLMF: Untitled Document

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29.10.2

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

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : 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}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),

►

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

►

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

… ►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}}

. …

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See accompanying text — Figure 29.13.5: $\mathit{uE}^{m}_{4}\left(x,0.1\right)$ for $-2K\leq x\leq 2K$ , $m=0,1,2$ . $K=1.61244\dotsc$ . Magnify

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10.35.2

e^{z\cos\theta}=I_{0}\left(z\right)+2\sum_{k=1}^{\infty}I_{k}\left(z\right)% \cos\left(k\theta\right),

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Symbols:: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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)+\dotsb,

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Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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)+\dots,

►

\sinh z=2I_{1}\left(z\right)+2I_{3}\left(z\right)+2I_{5}\left(z\right)+\dots.

… ►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

,

\operatorname{cn}\left(Kx,k\right)

,

\operatorname{dn}\left(Kx,k\right)

,

\operatorname{am}\left(Kx,k\right)

,

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. ►Curtis (1964b) tabulates

\operatorname{sn}\left(mK/n,k\right)

,

\operatorname{cn}\left(mK/n,k\right)

,

\operatorname{dn}\left(mK/n,k\right)

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

\operatorname{sn}\left(Kx,k\right)

,

\operatorname{cn}\left(Kx,k\right)

,

\operatorname{dn}\left(Kx,k\right)

for

k=\frac{1}{4},\frac{1}{2}

and

x=0(.1)4

to 7D. …

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29.14.2

\langle g,h\rangle=\int_{0}^{K}\!\!\int_{0}^{{K^{\prime}}}w(s,t)g(s,t)h(s,t)\,% \mathrm{d}t\,\mathrm{d}s,

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : 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)={\operatorname{sn}}^{2}\left(K+\mathrm{i}t,k\right)-{\operatorname{sn}}% ^{2}\left(s,k\right).

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : 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

\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),

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Symbols:: $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $K\left(\NVar{k}\right)$ : 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

\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),

ⓘ

Symbols:: $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $K\left(\NVar{k}\right)$ : 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

\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \,\mathrm{d}t\,\mathrm{d}s,

ⓘ

Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : 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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Table 22.4.1: Periods and poles of Jacobian elliptic functions.

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Periods	$z$ -Poles
Periods	$iK^{\prime}$	$K+iK^{\prime}$	$K$	$0$
…

► … ►

Table 22.4.2: Periods and zeros of Jacobian elliptic functions.

► ►►►

Periods	$z$ -Zeros
Periods	$0$	$K$	$K+iK^{\prime}$	$iK^{\prime}$
…

► ►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+2iK^{\prime}

,

2iK^{\prime}

. … ►Figure 22.4.2 depicts the fundamental unit cell in the

z

-plane, with vertices

\mbox{s}=0

,

\mbox{c}=K

,

\mbox{d}=K+iK^{\prime}

,

\mbox{n}=iK^{\prime}

. … ►This half-period will be plus or minus a member of the triple

{K,iK^{\prime},K+iK^{\prime}}

; the other two members of this triple are quarter periods of

\operatorname{pq}\left(z,k\right)

. …

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10.28.1

\mathscr{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),

ⓘ

Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\sin\NVar{z}$ : 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

\mathscr{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.

ⓘ

Symbols:: $\mathscr{W}$ : Wronskian, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

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14.10.2

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

ⓘ

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

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{% \mu}_{\nu}\left(x\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : 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}^{\mu}_{\nu}\left(x\right)

also satisfies (14.10.1)–(14.10.5). …In addition,

P^{\mu}_{\nu}\left(x\right)

and

Q^{\mu}_{\nu}\left(x\right)

satisfy (14.10.3)–(14.10.5).

… ►Assume

I_{\nu-1}\left(z\right)\neq 0

. … ►

10.33.1

\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,

z\neq 0

,

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.33 and Ch.10

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10.33.2

\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,

\nu\neq 0,-1,-2,\dotsc

.

ⓘ

Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.33 and Ch.10

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first kind

1—10 of 282 matching pages

1: 29.10 Lamé Functions with Imaginary Periods

2: 29.13 Graphics

3: 10.35 Generating Function and Associated Series

4: 22.21 Tables

5: 29.14 Orthogonality

6: 22.3 Graphics

7: 22.4 Periods, Poles, and Zeros

8: 10.28 Wronskians and Cross-Products

9: 14.10 Recurrence Relations and Derivatives

10: 10.33 Continued Fractions