DLMF: Untitled Document

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M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:: MathReview (E. Weber), ZentralBlatt
Referenced by:: §33.9(ii).
Encodings:: BibTeX

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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.

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

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D. E. Amos (1974) Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

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External Links:: ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt
Referenced by:: §10.74(v).
Encodings:: BibTeX

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D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.

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External Links:: FullText
Referenced by:: 1st item, B. Döring (1966).
Encodings:: BibTeX

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

K

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

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

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§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

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Döring (1966) tabulates all zeros of $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , ${H^{(1)}_{0}}\left(z\right)$ , ${H^{(1)}_{1}}\left(z\right)$ , that lie in the sector $|z|<158$ , $|\operatorname{ph}z|\leq\pi$ , to 10D. Some of the smaller zeros of $Y_{n}\left(z\right)$ and ${H^{(1)}_{n}}\left(z\right)$ for $n=2,3,4,5,15$ are also included.

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§10.75(iv) Integrals of Bessel Functions

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§10.75(v) Modified Bessel Functions and their Derivatives

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§10.75(vii) Integrals of Modified Bessel Functions

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B. Gabutti (1979) On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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External Links:: ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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External Links:: ISSN 0771-050X, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

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A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

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A. Gervois and H. Navelet (1986b) Some integrals involving three modified Bessel functions. II. J. Math. Phys. 27 (3), pp. 688–695.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

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J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii), 7th item, §10.77(ix).
Encodings:: BibTeX

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F. E. Relton (1965) Applied Bessel Functions. Dover Publications Inc., New York.

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Notes:: Reprint of original Blackie & Son Limited, London, 1946.
External Links:: MathReview, ZentralBlatt
Referenced by:: §10.73(iii).
Encodings:: BibTeX

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G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67
External Links:: ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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

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

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K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii).
Encodings:: BibTeX

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A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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V. Bezvoda, R. Farzan, K. Segeth, and G. Takó (1986) On numerical evaluation of integrals involving Bessel functions. Apl. Mat. 31 (5), pp. 396–410.

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External Links:: ISSN 0373-6725, MathReview (G. A. Evans), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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Yu. A. Brychkov and K. O. Geddes (2005) On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec. Funct. 16 (3), pp. 187–198.

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External Links:: ISSN 1065-2469, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.15, §10.38, §11.4(vi).
Encodings:: BibTeX

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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A. Sidi (1997) Computation of infinite integrals involving Bessel functions of arbitrary order by the

\overline{D}

-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:: ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt
Referenced by:: (9.7.16), (9.7.17), §9.7(iii), §9.7(iv).
Encodings:: BibTeX

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J. N. Newman (1984) Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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P. L. Kapitsa (1951b) The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §10.21(xiii).
Encodings:: BibTeX

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov and S. L. Skorokhodov (1985b) Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 26–36
External Links:: ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi), 4th item, 5th item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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B. G. Korenev (2002) Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:: Translated from the Russian by E. V. Pankratiev
External Links:: ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt
Referenced by:: §10.73(i), §10.73(i), §10.73(i).
Encodings:: BibTeX

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J. B. Campbell (1984) Determination of

\nu

-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

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Notes:: Double-precision Fortran, maximum accuracy 7S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 1st item.
Encodings:: BibTeX

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R. Cicchetti and A. Faraone (2004) Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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External Links:: ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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J. A. Cochran and J. N. Hoffspiegel (1970) Numerical techniques for finding

\nu

-zeros of Hankel functions. Math. Comp. 24 (110), pp. 413–422.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

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J. A. Cochran (1965) The zeros of Hankel functions as functions of their order. Numer. Math. 7 (3), pp. 238–250.

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External Links:: ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

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A. Cruz, J. Esparza, and J. Sesma (1991) Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

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	Open Source					With Book		Commercial
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10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)	✓	✓	✓	a	✓		✓	✓	✓	✓	✓
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

integrals%20of%20Bessel%20and%20Hankel%20functions

1—10 of 11 matching pages

1: Bibliography

2: 10.75 Tables

§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

§10.75(iv) Integrals of Bessel Functions

§10.75(v) Modified Bessel Functions and their Derivatives

§10.75(vii) Integrals of Modified Bessel Functions

3: Bibliography G

4: Bibliography R

5: Bibliography B

6: Bibliography S

7: Bibliography N

8: Bibliography K

9: Bibliography C

10: Software Index