DLMF: Untitled Document

… ►

2^{20}\beta_{5}=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{% 5}+1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.

… ►As

\gamma^{2}\to-\infty

, with

q=n+1

if

n-m

is even, or

q=n

if

n-m

is odd, we have …

…

… ►

See accompanying text — Figure 22.3.13: $\operatorname{sn}\left(x,k\right)$ for $k=1-e^{-n}$ , $n=0$ to 20, $-5\pi\leq x\leq 5\pi$ . Magnify 3D Help

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►

… ►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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•

Woodward and Woodward (1946) tabulates the real and imaginary parts of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ , $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ for $\Re z=-2.4(.2)2.4$ , $\Im z=-2.4(.2)0$ . Precision is 4D.

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•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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•

Corless et al. (1992) gives the real and imaginary parts of $\beta_{k}$ for $k=1(1)13$ ; 14S.

…

… ►

20.10.1

\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E1
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.31) and (20.4.5).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2

\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E2
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.32) and (20.4.4).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3

\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E3
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle|ix^{% -2}\right)\sim 1$ as $x\to 0+$ , which follows from (20.7.33) and (20.4.3).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

… ►Let

s

,

\ell

, and

\beta

be constants such that

\Re s>0

,

\ell>0

, and

\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell

. …

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

►

•

Zhang and Jin (1996, pp. 185–195) tabulates $J_{n}\left(x\right)$ , $J_{n}'\left(x\right)$ , $Y_{n}\left(x\right)$ , $Y_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1$ , 5, 10, 25, 50, 100, 9S; $J_{n+\alpha}\left(x\right)$ , $J_{n+\alpha}'\left(x\right)$ , $Y_{n+\alpha}\left(x\right)$ , $Y_{n+\alpha}'\left(x\right)$ , $n=0(1)5,10,30,50,100$ , $\alpha=\tfrac{1}{4},\tfrac{1}{3},\tfrac{1}{2},\tfrac{2}{3},\tfrac{3}{4}$ , $x=1,5,10,50$ , 8S; real and imaginary parts of $J_{n+\alpha}\left(z\right)$ , $J_{n+\alpha}'\left(z\right)$ , $Y_{n+\alpha}\left(z\right)$ , $Y_{n+\alpha}'\left(z\right)$ , $n=0(1)15,20(10)50,100$ , $\alpha=0,\tfrac{1}{2}$ , $z=4+2i$ , $20+10i$ , 8S.

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•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Zhang and Jin (1996, pp. 240–250) tabulates $I_{n}\left(x\right)$ , $I_{n}'\left(x\right)$ , $K_{n}\left(x\right)$ , $K_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1,5,10,25,50,100$ , 9S; $I_{n+\alpha}\left(x\right)$ , $I_{n+\alpha}'\left(x\right)$ , $K_{n+\alpha}\left(x\right)$ , $K_{n+\alpha}'\left(x\right)$ , $n=0(1)5$ , 10, 30, 50, 100, $\alpha=\tfrac{1}{4}$ , $\tfrac{1}{3}$ , $\tfrac{1}{2}$ , $\tfrac{2}{3}$ , $\tfrac{3}{4}$ , $x=1$ , 5, 10, 50, 8S; real and imaginary parts of $I_{n+\alpha}\left(z\right)$ , $I_{n+\alpha}'\left(z\right)$ , $K_{n+\alpha}\left(z\right)$ , $K_{n+\alpha}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $\alpha=0,\tfrac{1}{2}$ , $z=4+2i,20+10i$ , 8S.

… ►

•

Zhang and Jin (1996, pp. 296–305) tabulates $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , respectively.)

…

… ►

12.14.8

W\left(a,x\right)=W\left(a,0\right)w_{1}(a,x)+W'\left(a,0\right)w_{2}(a,x).

ⓘ

Symbols:: $W\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function, $x$ : real variable, $a$ : real or complex parameter, $w_{1}(a,x)$ : even solution and $w_{2}(a,x)$ : odd solution
A&S Ref:: 19.17.1 (modification of)
Permalink:: http://dlmf.nist.gov/12.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(v), §12.14 and Ch.12

►Here

w_{1}(a,x)

and

w_{2}(a,x)

are the even and odd solutions of (12.2.3): … ►

12.14.10

w_{2}(a,x)=\sum_{n=0}^{\infty}\beta_{n}(a)\frac{x^{2n+1}}{(2n+1)!},

ⓘ

Defines:: $w_{2}(a,x)$ : odd solution (locally)
Symbols:: $!$ : factorial (as in $n!$ ), $x$ : real variable, $n$ : nonnegative integer, $a$ : real or complex parameter and $\beta_{n}(a)$ : recursion
A&S Ref:: 19.16.2 (modification of)
Permalink:: http://dlmf.nist.gov/12.14.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(v), §12.14 and Ch.12

… ►The even and odd solutions of (12.2.3) (see §12.14(v)) are given by … ►The coefficients

c_{2r}

and

d_{2r}

are obtainable by equating real and imaginary parts in …

… ►

5.4.2

n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!!$ : double factorial and $n$ : nonnegative integer
Referenced by:: §5.4(i), §5.4(i)
Permalink:: http://dlmf.nist.gov/5.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.4(i), §5.4 and Ch.5

… ►

5.4.16

\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable
A&S Ref:: 6.3.11
Permalink:: http://dlmf.nist.gov/5.4.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §5.4(ii), §5.4 and Ch.5

►

5.4.17

\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable
A&S Ref:: 6.3.12
Permalink:: http://dlmf.nist.gov/5.4.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §5.4(ii), §5.4 and Ch.5

►

5.4.18

\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable
A&S Ref:: 6.3.13
Referenced by:: §5.4(ii)
Permalink:: http://dlmf.nist.gov/5.4.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §5.4(ii), §5.4 and Ch.5

…

… ►For the actual partitions (

\pi

) for

n=1(1)5

see Table 26.4.1. ►The integers whose sum is

n

are referred to as the parts in the partition. The example

\{1,1,1,2,4,4\}

has six parts, three of which equal 1. ►

Table 26.2.1: Partitions

p\left(n\right)

.

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

►

… ►

►

… ►The series also converges when

|z|=1

, provided that

\Re s>1

. … ►valid when

\Re s>0

and

\left|\operatorname{ph}\left(1-z\right)\right|<\pi

, or

\Re s>1

and

z=1

. … ►valid when

\Re s>0

,

\Im a>0

or

\Re s>1

,

\Im a=0

. …

odd%20part

21—30 of 400 matching pages

21: 30.9 Asymptotic Approximations and Expansions

22: 10.59 Integrals

23: 22.3 Graphics

24: 9.18 Tables

25: 20.10 Integrals

26: 10.75 Tables

27: 12.14 The Function $W\left(a,x\right)$

28: 5.4 Special Values and Extrema

29: 26.2 Basic Definitions

30: 25.12 Polylogarithms