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11: 10.18 Modulus and Phase Functions

… ►

$N_{\nu }^2\left(x\right){\varphi }_{\nu }^{\prime }\left(x\right)={\displaystyle \frac{2(x^2-{\nu }^2)}{\pi x^3}},$

… ►As $x\to \mathrm{\infty }$, with $\nu$ fixed and $\mu =4{\nu }^2$, … ►

10.18.20 $$-\frac{(2k-3)!!}{(2k)!!}\frac{(\mu -1)(\mu -9)\mathrm{\cdots }(\mu -{(2k-3)}^2)(\mu -(2k+1){(2k-1)}^2)}{{(2x)}^{2k}},$$ $k\ge 2$,

ⓘ

Symbols:

$!!$: double factorial, $k$: nonnegative integer and $x$: real variable

Referenced by:

§10.18(iii), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.18.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

… ►

10.18.21 $${\varphi }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu -\frac{1}{4}\right)\pi +\frac{\mu +3}{2(4x)}+\frac{{\mu }^2+46\mu -63}{6{(4x)}^3}+\frac{{\mu }^3+185{\mu }^2-2053\mu +1899}{5{(4x)}^5}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.31

Referenced by:

§10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

12: 13.10 Integrals

… ►

13.10.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt-t^2}{t}^{2b-2}𝐌(a,b,t^2)dt=\frac{1}{2}{\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(b-\frac{1}{2}\right)U(b-\frac{1}{2},a+\frac{1}{2},\frac{1}{4}{z}^2),$$ $\mathrm{\Re }b>\frac{1}{2}$, $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►

13.10.9 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{tz}{t}^{-a}U(a,b,y/t)dt=\frac{2z^{\frac{1}{2}(2a-b-1)}{y}^{\frac{1}{2}(1-b)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}{K}_{b-1}\left(2\sqrt{zy}\right),$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

… ►

13.10.12 $${\int }_0^{\mathrm{\infty }}\cos \left(2xt\right)𝐌(a,b,-t^2)dt=\frac{\sqrt{\pi }}{2\mathrm{\Gamma }\left(a\right)}{x}^{2a-1}{\mathrm{e}}^{-x^2}U(b-\frac{1}{2},a+\frac{1}{2},x^2),$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(iv), §13.10 and Ch.13

… ►

13.10.15 $${\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -b+2\right)}{\mathrm{\Gamma }\left(a\right)}{x}^{\frac{1}{2}\nu }U(\nu -b+2,\nu -a+2,x),$$ $x>0$, ,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §§1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). …

13: 4.17 Special Values and Limits

… ►

Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
…
$\pi /4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
…
$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
$3\pi /4$	$\frac{1}{2}\sqrt{2}$	$-\frac{1}{2}\sqrt{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
…

►

… ►

4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

14: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►If any lower argument in a $\mathit{6}j$ symbol is $0$, $\frac{1}{2}$, or $1$, then the $\mathit{6}j$ symbol has a simple algebraic form. … ►

34.5.5

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.6

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.7

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

… ►

34.5.13 $$E(j)={\left((j^2-{(j_2-j_3)}^2)({(j_2+j_3+1)}^2-j^2)(j^2-{(l_2-l_3)}^2)({(l_2+l_3+1)}^2-j^2)\right)}^{\frac{1}{2}}.$$

ⓘ

Defines:

$E(j)$ (locally)

Symbols:

$j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(iii), §34.5 and Ch.34

…

15: 19.28 Integrals of Elliptic Integrals

… ►

19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.2 $${\int }_0^1{t}^{\sigma -1}{R}_G(0,t,1)dt=\frac{\sigma }{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

ⓘ

Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Permalink:

http://dlmf.nist.gov/19.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

… ►

19.28.9 $${\int }_0^{\pi /2}{R}_F({\sin }^2\theta {\cos }^2\left(x+y\right),{\sin }^2\theta {\cos }^2\left(x-y\right),1)d\theta =R_F(0,{\cos }^{2}x,1)R_F(0,{\cos }^{2}y,1),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.10 $${\int }_0^{\mathrm{\infty }}{R}_F({(ac+bd)}^2,{(ad+bc)}^2,4abcd{\cosh }^{2}z)dz=\frac{1}{2}{R}_F(0,a^2,b^2)R_F(0,c^2,d^2),$$ $a,b,c,d>0$.

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

…

16: 29.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. … ►Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by … ►

$${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)=s_{\nu }^{2m+2}(k^2){\mathrm{Es}}_{\nu }^{2m+2}(z,k^2),$$

►where the positive factors $c_{\nu }^m(k^2)$ and $s_{\nu }^m(k^2)$ are determined by …

17: 4.30 Elementary Properties

… ►

Table 4.30.1: Hyperbolic functions: interrelations. …

► ►►►►►►►

	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
$\tanh \theta$	$a{(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	${(1-a^2)}^{1/2}$	$a^{-1}$
$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
$\mathrm{sech}\theta$	${(1+a^2)}^{-1/2}$	$a^{-1}$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a$	$a^{-1}{(a^2-1)}^{1/2}$
…

►

18: 29.15 Fourier Series and Chebyshev Series

… ►

Polynomial ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$

…

19: 22.9 Cyclic Identities

… ►

§22.9(ii) Typical Identities of Rank 2

… ►These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }m-1$; $n\to n+1\to n+2\to \mathrm{\cdots }p\to 1\to 2\to \mathrm{\cdots }n-1$. … ►

22.9.11 $${\left(d_{1,2}^{(2)}\right)}^2{d}_{2,2}^{(2)}\pm {\left(d_{2,2}^{(2)}\right)}^2{d}_{1,2}^{(2)}=k^{\prime }\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$$

ⓘ

Symbols:

$k^{\prime }$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

►

22.9.12 $$c_{1,2}^{(2)}{s}_{1,2}^{(2)}{d}_{2,2}^{(2)}+c_{2,2}^{(2)}{s}_{2,2}^{(2)}{d}_{1,2}^{(2)}=0.$$

ⓘ

Symbols:

$s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity and $d_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

… ►

22.9.21 $$k^2{c}_{1,2}^{(2)}{s}_{1,2}^{(2)}{c}_{2,2}^{(2)}{s}_{2,2}^{(2)}=k^{\prime }\left(1-{\left(s_{1,2}^{(2)}\right)}^2-{\left(s_{2,2}^{(2)}\right)}^2\right).$$

ⓘ

Symbols:

$k$: modulus, $k^{\prime }$: complementary modulus, $s_{m,p}^{(r)}$: cyclic quantity and $c_{m,p}^{(r)}$: cyclic quantity

Permalink:

http://dlmf.nist.gov/22.9.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

20: 27.2 Functions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►The $\varphi \left(n\right)$ numbers $a,a^2,\mathrm{\dots },a^{\varphi \left(n\right)}$ are relatively prime to $n$ and distinct (mod $n$). …It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. … ►

27.2.12

ⓘ

Defines:

$\mu \left(n\right)$: Möbius function

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a$: primitive root

A&S Ref:

24.3.1 I.A

Permalink:

http://dlmf.nist.gov/27.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

►