.1994%E5%B9%B4%E4%B8%96%E7%95%8C%E6%9D%AF%E5%B0%8F%E7%BB%84%E5%88%86%E7%BB%84_%E3%80%8E%E7%BD%91%E5%9D%80%3A68707.vip%E3%80%8F2018%E4%B8%96%E7%95%8C%E6%9D%AF%E7%9B%B4%E6%92%AD%E5%92%AA%E5%92%95_b5p6v3_2022%E5%B9%B411%E6%9C%8830%E6%97%A57%E6%97%B657%E5%88%8610%E7%A7%92_z9fzftt9n.cc

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11: Bibliography M

… ►

R. C. McCann (1977) Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

J. N. Merner (1962) Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

ⓘ

Notes:: Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

E. W. Montroll (1964) Lattice Statistics. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), University of California Engineering and Physical Sciences Extension Series, pp. 96–143.

ⓘ

Notes:: Reprinted by Robert E. Krieger Publishing Co. in 1981.
External Links:: ZentralBlatt
Referenced by:: §26.20.
Encodings:: BibTeX

…

12: 5.13 Integrals

… ►

5.13.1

{\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\Gamma\left% (b-s\right)z^{-s}\,\mathrm{d}s=\frac{\Gamma\left(a+b\right)z^{a}}{(1+z)^{a+b}}},

\Re\left(a+b\right)>0

-\Re a<c<\Re b

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\operatorname{ph}$ : phase, $\Re$ : real part, $s$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13, §5.13, §5.19(ii)
Permalink:: http://dlmf.nist.gov/5.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13 and Ch.5

►

5.13.2

{\frac{1}{2\pi}\int_{-\infty}^{\infty}|\Gamma\left(a+it\right)|^{2}e^{(2b-\pi)% t}\,\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b)^{2a}}},

a>0

0<b<\pi

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13 and Ch.5

… ►

5.13.3

\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a+it\right)\Gamma\left(b+it% \right)\Gamma\left(c-it\right)\Gamma\left(d-it\right)\,\mathrm{d}t=\frac{% \Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(% b+d\right)}{\Gamma\left(a+b+c+d\right)},

\Re a,\Re b,\Re c,\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13, §5.13 and Ch.5

… ►

5.13.4

\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{\Gamma\left(a+t\right)\Gamma\left(% b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t\right)}=\frac{\Gamma\left(a+b+% c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma\left(a+d-1\right)\Gamma\left(b+c-% 1\right)\Gamma\left(b+d-1\right)},

\Re\left(a+b+c+d\right)>3

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13, §5.13 and Ch.5

… ►

5.13.5

\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\Gamma\left(a_{k}+it% \right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2it\right)\Gamma\left(-2it% \right)}\,\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}\Gamma\left(a_{j}+a_{k}% \right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},

\Re\left(a_{k}\right)>0

k=1,2,3,4

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $j$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §5.13, §5.13 and Ch.5

…

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

34.5.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{% \frac{1}{2}}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}+\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_% {3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}-\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j_{2}(2j_{2}+1)2j_{3}(2j_{3}+% 1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.4

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1% )}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.5

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-% 2j_{3}+1)}{2j_{2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac% {1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ 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

…

14: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …where

p_{1},p_{2},\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

. … ►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. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. …

15: 7.14 Integrals

… ►

7.14.1

\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(bt\right)\,\mathrm{d}t={% \frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{2a}\left(1-e^{-(a/b)^{% 2}}\right)},

a\in\mathbb{C}

|\operatorname{ph}b|<\tfrac{1}{4}\pi

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{ph}$ : phase
A&S Ref:: 7.4.18 (in different form)
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

… ►

7.14.2

\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),

\Re a>0

|\operatorname{ph}b|<\tfrac{1}{4}\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.17
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

… ►

7.14.5

\int_{0}^{\infty}e^{-at}C\left(t\right)\,\mathrm{d}t=\frac{1}{a}\mathrm{f}% \left(\frac{a}{\pi}\right),

\Re a>0

ⓘ

Symbols:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.27 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

… ►

7.14.7

\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\,\mathrm{d}t=\frac% {(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},

\Re a>0

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.29 in modified form
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

… ►For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2000, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8). …

16: Bibliography F

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §5.24(ii).
Encodings:: BibTeX

… ►

V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §9.1, Notations U, Notations V.
Encodings:: BibTeX

… ►

C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

ⓘ

External Links:: Document
Referenced by:: §36.14(iv).
Encodings:: BibTeX

… ►

B. R. Frieden (1971) Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

ⓘ

Referenced by:: §30.15(i), §30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).
Encodings:: BibTeX

…

17: Bibliography B

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).
Encodings:: BibTeX

… ►

M. V. Berry (1975) Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.

ⓘ

External Links:: Document, MathReview
Referenced by:: §36.14(iii).
Encodings:: BibTeX

… ►

R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt
Referenced by:: §18.35(iii).
Encodings:: BibTeX

… ►

J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M. A. Shokrollahi (2001) Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31 (1-2), pp. 89–96.

ⓘ

Notes:: Computational algebra and number theory (Milwaukee, WI, 1996)
External Links:: ISSN 0747-7171, Document, MathReview (Xavier-François Roblot), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

18: 9.8 Modulus and Phase

… ►In terms of Bessel functions, and with

\xi=\tfrac{2}{3}|x|^{3/2}

, … ►

9.8.20

{M}^{2}\left(x\right)\sim\frac{1}{\pi(-x)^{1/2}}\sum_{k=0}^{\infty}\frac{1% \cdot 3\cdot 5\cdots(6k-1)}{k!(96)^{k}}\frac{1}{x^{3k}},

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $M\left(\NVar{z}\right)$ : Airy modulus function and $x$ : real variable
Proof sketch:: Combine (9.8.9)–(9.8.12) with §10.18(iii)
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §9.8(iv), §9.8 and Ch.9

►

9.8.21

{N}^{2}\left(x\right)\sim\frac{(-x)^{1/2}}{\pi}\sum_{k=0}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(6k-1)}{k!(96)^{k}}\frac{1+6k}{1-6k}\frac{1}{x^{3k}},

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $N\left(\NVar{z}\right)$ : Airy modulus function and $x$ : real variable
Proof sketch:: Combine (9.8.9)–(9.8.12) with §10.18(iii)
Referenced by:: §9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §9.8(iv), §9.8 and Ch.9

►

9.8.22

\theta\left(x\right)\sim\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1+\frac{5}{32% }\frac{1}{x^{3}}+\frac{1105}{6144}\frac{1}{x^{6}}+\frac{82825}{65536}\frac{1}{% x^{9}}+\frac{12820\;31525}{587\;20256}\frac{1}{x^{12}}+\cdots\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta\left(\NVar{z}\right)$ : Airy phase function and $x$ : real variable
Proof sketch:: Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).
Referenced by:: §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §9.8(iv), §9.8 and Ch.9

►

9.8.23

\phi\left(x\right)\sim-\frac{\pi}{4}+\frac{2}{3}(-x)^{3/2}\left(1-\frac{7}{32}% \frac{1}{x^{3}}-\frac{1463}{6144}\frac{1}{x^{6}}-\frac{4\;95271}{3\;27680}% \frac{1}{x^{9}}-\frac{2065\;30429}{83\;88608}\frac{1}{x^{12}}-\cdots\right).

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\phi\left(\NVar{z}\right)$ : Airy phase function and $x$ : real variable
Proof sketch:: Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).
Referenced by:: §9.8(iv), §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)
Permalink:: http://dlmf.nist.gov/9.8.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §9.8(iv), §9.8 and Ch.9

…

19: 4.40 Integrals

… ►

4.40.7

\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{\sinh x}\,\mathrm{d}x=\tfrac% {1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1}{a},

a\neq 0

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.8

\int_{0}^{\infty}\frac{\sinh\left(ax\right)}{\sinh\left(\pi x\right)}\,\mathrm% {d}x=\tfrac{1}{2}\tan\left(\tfrac{1}{2}a\right),

-\pi<a<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\tan\NVar{z}$ : tangent function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.9

\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh\left(\tfrac{1}{2}x\right)% \right)^{2}}\,\mathrm{d}x=\frac{4\pi a}{\sin\left(\pi a\right)},

-1<a<1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.10

{\int_{0}^{\infty}\frac{\tanh\left(ax\right)-\tanh\left(bx\right)}{x}\,\mathrm% {d}x=\ln\left(\frac{a}{b}\right)},

a>0

b>0

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

… ►Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).

20: 9.9 Zeros

… ►They are denoted by

a_{k}

a^{\prime}_{k}

b_{k}

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. … ►

9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

… ►

9.9.21

W(t)\sim\pi^{-1/2}t^{-1/6}\left(1-\frac{7}{96}t^{-2}+\frac{1673}{6144}t^{-4}-% \frac{843\;94709}{265\;42080}t^{-6}+\frac{78\;02771\;35421}{1\;01921\;58720}t^% {-8}-\frac{20444\;90510\;51945}{6\;52298\;15808}t^{-10}+\cdots\right).

ⓘ

Defines:: $W$ : expansion (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion and $\pi$ : the ratio of the circumference of a circle to its diameter
Source:: Olver (1954, (A 19))
Permalink:: http://dlmf.nist.gov/9.9.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

… ►For error bounds for the asymptotic expansions of

a_{k}

b_{k}

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …