fractional integrals

(0.003 seconds)

31—40 of 64 matching pages

31: 14.17 Integrals

… ►

14.17.7

\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{P}^{-m}_{n}\left(x\right)% \,\mathrm{d}x=\frac{(-1)^{m}}{l+\frac{1}{2}}\delta_{l,n},

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E7
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right and $(-1)^{m}$ has been moved to the numerator of the fraction.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

… ►

14.17.9

\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right)\mathsf{P}^{-m}_{n}\left(x% \right)}{1-x^{2}}\,\mathrm{d}x=\frac{(-1)^{l}}{l}\delta_{l,m},

l>0

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
Referenced by:: §14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)
Permalink:: http://dlmf.nist.gov/14.17.E9
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right and $(-1)^{l}$ has been moved to the numerator of the fraction.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

…

32: Bibliography T

… ►

G. ’t Hooft and M. Veltman (1979) Scalar one-loop integrals. Nuclear Phys. B 153 (3-4), pp. 365–401.

ⓘ

External Links:: ISSN 0029-5582, Document, MathReview (L. O’Raifeartaigh)
Referenced by:: §25.12(i).
Encodings:: BibTeX

… ►

I. J. Thompson and A. R. Barnett (1985) COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item, §33.23(vi), 3rd item.
Encodings:: BibTeX

… ►

I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

ⓘ

External Links:: Document, Code, ZentralBlatt
Referenced by:: §33.23(vi), I. J. Thompson and A. R. Barnett (1985).
Encodings:: BibTeX

… ►

J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

ⓘ

External Links:: FullText
Referenced by:: §25.21(vii).
Encodings:: BibTeX

… ►

J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

ⓘ

External Links:: MathReview (H. Bückner), ZentralBlatt
Referenced by:: §6.18(i).
Encodings:: BibTeX

…

33: Bibliography

… ►

N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

ⓘ

Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

… ►

J. R. Albright and E. P. Gavathas (1986) Integrals involving Airy functions. J. Phys. A 19 (13), pp. 2663–2665.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: (9.11.11), (9.11.12), (9.11.13), (9.11.14), §9.11(iv).
Encodings:: BibTeX

… ►

Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

ⓘ

External Links:: Document
Referenced by:: §10.73(iv), §8.24(iii).
Encodings:: BibTeX

… ►

D. E. Amos (1980b) Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.

ⓘ

External Links:: ISSN 0098-3500, Document, MathReview (M. M. Chawla), ZentralBlatt
Referenced by:: §8.25(v).
Encodings:: BibTeX

… ►

R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.

ⓘ

External Links:: ISSN 0065-9266, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

…

34: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where

x, y, z

may be permuted. … ►The general lemniscatic case is … ►

35: Bibliography R

… ►

Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).
Encodings:: BibTeX

►

W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: (9.11.16), (9.11.17), §9.11(iii), §9.11(v).
Encodings:: BibTeX

… ►

M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt
Referenced by:: §10.23(ii).
Encodings:: BibTeX

… ►

G. B. Rybicki (1989) Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

ⓘ

Referenced by:: §7.25(vi).
Encodings:: BibTeX

…

36: 8.25 Methods of Computation

… ►

§8.25(iv) Continued Fractions

►The computation of

\gamma\left(a,z\right)

and

\Gamma\left(a,z\right)

by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). … ►Stable recursive schemes for the computation of

E_{p}\left(x\right)

are described in Miller (1960) for

x>0

and integer

p

. …

37: 20.11 Generalizations and Analogs

… ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus

k

of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in

q

-series via (20.9.1). …

38: Bibliography C

… ►

B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

… ►

A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

ⓘ

Notes:: Fortran code.
External Links:: ISSN 0098-3500, Document
Referenced by:: §10.77(ix).
Encodings:: BibTeX

… ►

A. R. Curtis (1964b) Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.

ⓘ

External Links:: ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt
Referenced by:: §22.20(vii), §22.21.
Encodings:: BibTeX

►

A. Cuyt, V. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, and C. Bonan-Hamada (2007) Handbook of Continued Fractions for Special Functions. Kluwer Academic Publishers Group, Dordrecht.

ⓘ

Referenced by:: Profile Annie A. M. Cuyt.
Encodings:: BibTeX

►

A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, and W. B. Jones (2008) Handbook of Continued Fractions for Special Functions. Springer, New York.

ⓘ

External Links:: ISBN 978-1-4020-6948-2, MathReview Entry, ZentralBlatt
Referenced by:: §10.10, §10.33, §10.55, §10.74(v), §12.6, §13.17, §13.5, §15.7, §16.4(iv), §17.6(vi), §17.7(ii), §18.13, §3.10(ii), §3.10(iii), §4.25, §4.39, §4.9(i), §4.9(ii), §4.9(iii), §5.10, §5.15, §6.9, §7.18(v), §7.22(i), §7.9, §8.17(v), §8.19(vii), §8.9.
Encodings:: BibTeX

…

39: 11.15 Approximations

… ►

•

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{L}_{n}\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , $x\geq 8$ , for $n=0,1$ ; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ , $0\leq\left|x\right|\leq 8$ , $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ , $x\geq 8$ ; the coefficients are to 20D.

… ►

•

Newman (1984) gives polynomial approximations for $\mathbf{H}_{n}\left(x\right)$ for $n=0,1$ , $0\leq x\leq 3$ , and rational-fraction approximations for $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ for $n=0,1$ , $x\geq 3$ . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.

40: 18.2 General Orthogonal Polynomials

… ►

18.2.1

\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=0,

n\neq m

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : interval endpoint, $b$ : interval endpoint and $x$ : real variable
A&S Ref:: 22.1.1
Referenced by:: §18.2(i), §18.2(iii), §18.2(viii), §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

… ►More generally than (18.2.1)–(18.2.3),

w(x)\,\mathrm{d}x

may be replaced in (18.2.1) by

\,\mathrm{d}\mu(x)

, where the measure

\mu

is the Lebesgue–Stieltjes measure

\mu_{\alpha}

corresponding to a bounded nondecreasing function

\alpha

on the closure of

(a,b)

with an infinite number of points of increase, and such that

\int_{a}^{b}|x|^{n}\,\mathrm{d}\mu(x)<\infty

for all

n

. … ►

18.2.24

\lambda_{n}={h_{n}^{-1}}\int_{a}^{b}f(x)p_{n}(x)w(x)\,\mathrm{d}x

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$
Referenced by:: §18.18(i)
Permalink:: http://dlmf.nist.gov/18.2.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §18.2(viii), §18.2 and Ch.18

… ►

§18.2(x) Orthogonal Polynomials and Continued Fractions

… ►Using the terminology of §1.12(ii), the

n

-th approximant of the continued fraction …