DLMF: Untitled Document

§20.10 Integrals

… ►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

. Then ►

20.10.4

\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i% \pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=% -\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(% \ell\sqrt{s}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta 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, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.10(ii), §20.10 and Ch.20

… ►For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

§8.26(iii) Incomplete Beta Functions

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

W. A. Al-Salam and M. E. H. Ismail (1994) A

q

-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

-,

\beta

- and

\gamma

-Spectroscopy:

3j

-,

6j

-,

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

ⓘ

External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

…

… ► ►►►►►►►

	Open Source											With Book			Commercial
…
5.24(v) $\mathrm{B}\left(a,b\right)$ , $a,b\in\mathbb{R}$	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
5.24(vi) $\mathrm{B}\left(a,b\right)$ , $a,b\in\mathbb{C}$	✓	✓		✓					✓	a	✓				✓	✓	✓	a
…
19 Elliptic Integrals
…
20 Theta Functions
…
24 Bernoulli and Euler Polynomials
…

…

… ►

P. I. Pastro (1985) Orthogonal polynomials and some

q

-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.33(iv), §18.33(v).
Encodings:: BibTeX

… ►

K. Pearson (Ed.) (1965) Tables of the Incomplete

\Gamma

-function. Biometrika Office, Cambridge University Press, Cambridge.

ⓘ

Referenced by:: 3rd item.
Encodings:: BibTeX

►

K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.

ⓘ

Notes:: Includes Basic code.
External Links:: Document
Referenced by:: §8.28(iv).
Encodings:: BibTeX

… ►

R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

…

… ►

Equation (18.12.2)

18.12.2

{{}_{0}{\mathbf{F}}_{1}}\left({-\atop\alpha+1};\frac{(x-1)z}{2}\right)\*{{}_{0% }{\mathbf{F}}_{1}}\left({-\atop\beta+1};\frac{(x+1)z}{2}\right)=\left(\tfrac{1% }{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*% \left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)% z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% \Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n}

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${{}_{0}{\mathbf{F}}_{1}}$ functions.

… ►

Equations (31.3.10), (31.3.11)

31.3.10

z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\alpha\left(\beta-% \epsilon\right)-\frac{\alpha}{a}\left(\beta-\delta\right);\alpha,\alpha-\gamma% +1,\alpha-\beta+1,\delta;\frac{1}{z}\right)

31.3.11

z^{-\beta}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\beta\left(\alpha-% \epsilon\right)-\frac{\beta}{a}\left(\alpha-\delta\right);\beta,\beta-\gamma+1% ,\beta-\alpha+1,\delta;\frac{1}{z}\right)

In both equations, the second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.

… ►

Paragraph Case III:

V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}

(in §22.19(ii))

Two corrections have been made in this paragraph. First, the correct range of the initial displacement $a$ is $\sqrt{1/\beta}\leq|a|<\sqrt{2/\beta}$ . Previously it was $\sqrt{1/\beta}\leq|a|\leq\sqrt{2/\beta}$ . Second, the correct period of the oscillations is $2\!K\left(k\right)/\sqrt{\eta}$ . Previously it was given incorrectly as $4\!K\left(k\right)/\sqrt{\eta}$ .

Reported 2014-05-02 by Svante Janson.

… ►

Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

Equation (17.13.3)

17.13.3

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)}

Originally the differential was identified incorrectly as $\,{\mathrm{d}}_{q}t$ ; the correct differential is $\,\mathrm{d}t$ .

Reported 2011-04-08.

…

… ►

M. Carmignani and A. Tortorici Macaluso (1985) Calcolo delle funzioni speciali

\Gamma(x)

,

\log\Gamma(x)

,

\beta(x,y)

,

\operatorname{erf}(x)

,

\operatorname{erfc}(x)

alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

ⓘ

Notes:: Translated title: Computation of the special functions $\Gamma(x),\;{\rm log}\,\Gamma(x),\;\beta(x,y),\;{\rm erf}\,(x),\;{\rm erfc}\,(x)$ to a high degree of precision
External Links:: MathReview, ZentralBlatt
Referenced by:: §5.24(ii).
Encodings:: BibTeX

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

Th. Clausen (1828) Über die Fälle, wenn die Reihe von der Form

y=1+\frac{\alpha}{1}\cdot\frac{\beta}{\gamma}x+\frac{\alpha\cdot\alpha+1}{1% \cdot 2}\cdot\frac{\beta\cdot\beta+1}{\gamma\cdot\gamma+1}x^{2}+

etc. ein Quadrat von der Form

z=1+\frac{\alpha^{\prime}}{1}\cdot\frac{\beta^{\prime}}{\gamma^{\prime}}\cdot% \frac{\delta^{\prime}}{\epsilon^{\prime}}x+\frac{\alpha^{\prime}\cdot\alpha^{% \prime}+1}{1\cdot 2}\cdot\frac{\beta^{\prime}\cdot\beta^{\prime}+1}{\gamma^{% \prime}\cdot\gamma^{\prime}+1}\cdot\frac{\delta^{\prime}\cdot\delta^{\prime}+1% }{\epsilon^{\prime}\cdot\epsilon^{\prime}+1}x^{2}+

etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

ⓘ

External Links:: Link, MathReview Entry, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

…

… ►

C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

ⓘ

Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

… ►

A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 14D.
External Links:: ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

… ►

J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

ⓘ

External Links:: ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt
Referenced by:: §8.17(i).
Encodings:: BibTeX

…

Euler%20beta%20integral

8 matching pages

1: 20.10 Integrals

§20.10 Integrals

2: 8.26 Tables

§8.26(iii) Incomplete Beta Functions

3: Bibliography

4: Software Index

5: Bibliography P

6: Errata

7: Bibliography C

8: Bibliography D