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11: 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

… ►For compendia of integrals of gamma functions see Apelblat (1983, pp. 124–127 and 129–130), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, pp. 644–652), Oberhettinger (1974, pp. 191–204), Oberhettinger and Badii (1973, pp. 307–316), Prudnikov et al. (1986b, pp. 57–64), Prudnikov et al. (1992a, pp. 127–130), and Prudnikov et al. (1992b, pp. 113–123). …

12: 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

…

13: 27.2 Functions

… ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►

27.2.9

d\left(n\right)=\sum_{d\mathbin{|}n}1

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.3 I.A
Permalink:: http://dlmf.nist.gov/27.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►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

. …

14: 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). …

15: 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

…

16: Bibliography B

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

… ►

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

…

17: 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

…

18: 4.40 Integrals

… ►

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).

19: 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. ►For the distribution in

\mathbb{C}

of the zeros of

\operatorname{Ai}'\left(z\right)-\sigma\operatorname{Ai}\left(z\right)

, where

\sigma

is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►

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). …

20: Bibliography J

… ►

L. Jacobsen, W. B. Jones, and H. Waadeland (1986) Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

ⓘ

External Links:: MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §8.25(iv).
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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N. Joshi and A. V. Kitaev (2005) The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:: ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt
Referenced by:: §32.11(i).
Encodings:: BibTeX

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N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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External Links:: ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt
Referenced by:: §32.11(i), §32.12(ii).
Encodings:: BibTeX

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G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).

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External Links:: FullText, ZentralBlatt
Referenced by:: §3.8(viii).
Encodings:: BibTeX