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11: Software Index

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	Open Source													With Book						Commercial
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10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)	✓	✓	✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM, NMS
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10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions	✓	✓	✓									a	✓	✓			✓	✓	✓	✓	✓	✓	✓
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12: Bibliography V

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Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§28.36(iii).

Encodings:

BibTeX

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13: 24.16 Generalizations

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Polynomials and Numbers of Integer Order

… ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

14: Bibliography S

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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Algorithm 721: MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Software 19 (3), pp. 391–406.

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Notes:

Double-precision Fortran, maximum accuracy 18D

External Links:

ISSN 0098-3500, Document, GAMS (module 721; package TOMS), ZentralBlatt

Referenced by:

3rd item, 5th item.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

15: 11.2 Definitions

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11.2.1 $${𝐇}_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu +1}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)},$$

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Defines:

${𝐇}_{\nu }\left(z\right)$: Struve function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $n$: integer order

A&S Ref:

12.1.3

Referenced by:

§11.10(vi), §11.13(ii), §11.2(i), §11.4(iii), §11.6(ii), §11.6(ii)

Permalink:

http://dlmf.nist.gov/11.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

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11.2.2 $${𝐋}_{\nu }\left(z\right)=-\mathrm{i}{\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}\nu }{𝐇}_{\nu }\left(\mathrm{i}z\right)={(\frac{1}{2}z)}^{\nu +1}\sum _{n=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)}.$$

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Defines:

${𝐋}_{\nu }\left(z\right)$: modified Struve function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: real or complex order and $n$: integer order

A&S Ref:

12.2.1

Referenced by:

§11.13(ii), §11.2(i), §11.4(i), §11.4(iii), §11.5(ii), §11.5(i), §11.6(ii), §11.6(ii)

Permalink:

http://dlmf.nist.gov/11.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

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16: 11.10 Anger–Weber Functions

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11.10.10 $$S_1(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +1\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Referenced by:

(11.11.5), (11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

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11.10.11 $$S_2(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k+1}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +\frac{3}{2}\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +\frac{3}{2}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Referenced by:

§11.10(vi), (11.11.5), (11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

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11.10.22 $${𝐄}_n\left(z\right)=-{𝐇}_n\left(z\right)+\frac{1}{\pi }\sum _{k=0}^{m_1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+\frac{1}{2}-k\right)}{(\frac{1}{2}z)}^{n-2k-1},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.6 (with upper limit corrected in later printings.)

Referenced by:

§11.10(vi), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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11.10.23 $${𝐄}_{-n}\left(z\right)=-{𝐇}_{-n}\left(z\right)+\frac{{(-1)}^{n+1}}{\pi }\sum _{k=0}^{m_2}\frac{\mathrm{\Gamma }\left(n-k-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\frac{3}{2}\right)}{(\frac{1}{2}z)}^{-n+2k+1},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.7 (with upper limit corrected.)

Referenced by:

§11.10(vi), Ch.11

Permalink:

http://dlmf.nist.gov/11.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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11.10.29 $${𝐉}_n\left(z\right)=J_n\left(z\right),$$ $n\in \mathbb{Z}$.

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $\mathbb{Z}$: set of all integers, $z$: complex variable and $n$: integer order

A&S Ref:

12.3.2

Referenced by:

§11.10(vii), §11.11(ii)

Permalink:

http://dlmf.nist.gov/11.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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17: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

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18: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

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19: Bibliography P

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Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.

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External Links:

ISSN 0044-1899, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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20: Bibliography B

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Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Bessel Functions. Part II: Functions of Positive Integer Order. British Association for the Advancement of Science, Mathematical Tables, Volume 10, Cambridge University Press, Cambridge.

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Notes:

Reprinted, with corrections, in 1960.

External Links:

ISBN 0-521-04321-2, MathReview (R. C. Archibald), ZentralBlatt

Referenced by:

§10.18(iii), §10.19(ii), §10.21(vi), §10.41(ii), 2nd item, 2nd item, §3.6(iii).

Encodings:

BibTeX

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Algorithm 21: Bessel function for a set of integer orders. Comm. ACM 3 (11), pp. 600.

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Notes:

Includes Algol code, minimum accuracy 5S. For certification see same journal 8(1965), p. 219.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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