DLMF: Untitled Document

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B_{2n+1}=0

,

►

(-1)^{n+1}B_{2n}>0

,

n=1,2,\dots

.

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E_{2n+1}=0

,

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(-1)^{n}E_{2n}>0

.

… ►

\widetilde{E}_{n}\left(x\right)=E_{n}\left(x\right)

,

0\leq x<1

,

…

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•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

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•

Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

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•

Agrest et al. (1982) tabulates $\int_{0}^{x}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ and $e^{-x}\int_{0}^{x}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

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•

Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

§20.4 Values at $z$ = 0

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20.4.1

\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4 and Ch.20

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20.4.6

\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
A&S Ref:: 16.28.6
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

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20.4.7

\theta_{1}''\left(0,q\right)=\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(% 0,q\right)=\theta_{4}'''\left(0,q\right)=0.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.12

\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(ii), §20.4 and Ch.20

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Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

.

► ►►►►

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
…
$\cosh z$	$1$	$0$	$-1$	$0$	$\infty$
…
$\coth z$	$\infty$	$0$	$\infty$	$0$	$1$

► ►

4.31.1

\lim_{z\to 0}\frac{\sinh z}{z}=1,

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.2

\lim_{z\to 0}\frac{\tanh z}{z}=1,

ⓘ

Symbols:: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.3

\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

… ►

32.4.10

zu_{0}^{\prime}=\theta_{\infty}u_{0}-zv_{0}v_{1},

ⓘ

Symbols:: $z$ : real and $\theta_{\infty}$ : constant
Referenced by:: §32.4(iv)
Permalink:: http://dlmf.nist.gov/32.4.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

… ►

32.4.12

zv_{0}^{\prime}=2v_{0}u_{1}v_{1}+v_{0}+(u_{0}(2v_{0}-z)/v_{1}),

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

… ►If

w=-u_{0}/(v_{0}v_{1})

, then …where ►

32.4.16

\theta_{0}=\frac{4v_{0}}{z}\left(\theta_{\infty}\left(1-\frac{z}{4v_{0}}\right% )+\frac{z-2v_{0}}{2v_{0}v_{1}}u_{0}+u_{1}v_{1}\right).

ⓘ

Symbols:: $z$ : real, $\theta_{\infty}$ : constant and $\theta_{0}$
Permalink:: http://dlmf.nist.gov/32.4.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §32.4(iv), §32.4 and Ch.32

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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; $\mathsf{P}_{n}'\left(x\right)$ for $n=1(1)4,9,10$ , $x=0(.01)1$ , 5–7D; $\mathsf{Q}_{n}\left(x\right)$ and $\mathsf{Q}_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 6–8D; $P_{n}\left(x\right)$ and $P_{n}'\left(x\right)$ for $n=0(1)5,9,10$ , $x=1(.2)10$ , 6S; $Q_{n}\left(x\right)$ and $Q_{n}'\left(x\right)$ 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; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n-m=0(1)2$ , $n=10$ , $x=0,0.5$ , 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n=0(1)2,10$ , 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ 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 $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$ , $\theta=10^{\circ},30^{\circ},150^{\circ}$ .

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•

Belousov (1962) tabulates $\mathsf{P}^{m}_{n}\left(\cos\theta\right)$ (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 $-1<x<1$ .

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•

Žurina and Karmazina (1963) tabulates the conical functions $\mathsf{P}^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$ , $x=-0.9(.1)0.9$ , 7S; $P^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ 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 $-1<x<1$ .

…

… ►

15.15.1

\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(i)
Permalink:: http://dlmf.nist.gov/15.15.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §15.15 and Ch.15

►Here

z_{0}

(

{\neq 0}

) is an arbitrary complex constant and the expansion converges when

|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)

. …

… ►

•

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{L}_{n}\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , $x\geq 8$ , for $n=0,1$ ; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ , $0\leq\left|x\right|\leq 8$ , $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ , $x\geq 8$ ; the coefficients are to 20D.

►

•

MacLeod (1993) gives Chebyshev-series expansions for $\mathbf{L}_{0}\left(x\right)$ , $\mathbf{L}_{1}\left(x\right)$ , $0\leq x\leq 16$ , and $I_{0}\left(x\right)-\mathbf{L}_{0}\left(x\right)$ , $I_{1}\left(x\right)-\mathbf{L}_{1}\left(x\right)$ , $x\geq 16$ ; the coefficients are to 20D.

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•

Newman (1984) gives polynomial approximations for $\mathbf{H}_{n}\left(x\right)$ for $n=0,1$ , $0\leq x\leq 3$ , and rational-fraction approximations for $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ for $n=0,1$ , $x\geq 3$ . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

… ►

See accompanying text — Figure 28.3.1: $\operatorname{ce}_{2n}\left(x,1\right)$ for $0\leq x\leq\pi/2$ , $n=0,1,2,3$ . Magnify

►

… ►

►

… ►

…

… ►If

\gamma=0

and

\alpha\delta\neq 0

, then set

\alpha=1

and

\delta=-1

, without loss of generality. …Similar results hold for

\mbox{P}_{\mbox{\scriptsize III}}

with

\delta=0

and

\beta\gamma\neq 0

. … ►Let

W_{0}=W(z;\alpha_{0},\beta_{0},\gamma_{0},-\tfrac{1}{2})

and

W_{1}=W(z;\alpha_{1},\beta_{1},\gamma_{1},-\tfrac{1}{2})

be solutions of

\mbox{P}_{\mbox{\scriptsize V}}

, where …and assume

\Phi\neq 0

. … ►for

j=0,1,2,\infty

, where …

Appell%0Afunctions

21—30 of 701 matching pages

21: 24.2 Definitions and Generating Functions

22: 11.14 Tables

23: 20.4 Values at $z$ = 0

§20.4 Values at $z$ = 0

24: 4.31 Special Values and Limits

25: 32.4 Isomonodromy Problems

26: 14.33 Tables

27: 15.15 Sums

28: 11.15 Approximations

29: 28.3 Graphics

30: 32.7 Bäcklund Transformations

Appell%0Afunctions

21—30 of 701 matching pages

21: 24.2 Definitions and Generating Functions

22: 11.14 Tables

23: 20.4 Values at z = 0

§20.4 Values at z = 0

24: 4.31 Special Values and Limits

25: 32.4 Isomonodromy Problems

26: 14.33 Tables

27: 15.15 Sums

28: 11.15 Approximations

29: 28.3 Graphics

30: 32.7 Bäcklund Transformations

23: 20.4 Values at $z$ = 0

§20.4 Values at $z$ = 0