of integer order

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1: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

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2: 14.6 Integer Order

§14.6 Integer Order

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§14.6(i) Nonnegative Integer Orders

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§14.6(ii) Negative Integer Orders

… ►For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …

3: 11.1 Special Notation

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$x$	real variable.
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$n$	integer order.
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4: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions

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5: 14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

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§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

6: 11.4 Basic Properties

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§11.4(i) Half-Integer Orders

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11.4.1 $${𝐊}_{n+\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$

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Symbols:

${𝐊}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: integer order

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.2 $${𝐋}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{{(-1)}^m(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.3 $${𝐇}_{-n-\frac{1}{2}}\left(z\right)={(-1)}^n{J}_{n+\frac{1}{2}}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.1.15

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.4 $${𝐋}_{-n-\frac{1}{2}}\left(z\right)=I_{n+\frac{1}{2}}\left(z\right).$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.2.10

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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7: 14.34 Software

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Olver and Smith (1983). Integer order. Fortran.

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8: Philip J. Davis

… ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …

9: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. …

10: Bibliography

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Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

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External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.

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Notes:

Double-precision C++, minimum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 804; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Spherical Bessel functions $j_n$ and $y_n$ of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Notes:

Single-precision Fortran, maximum accuracy 10S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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