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31: Bruce R. Miller

… ►There, he carried out research in non-linear dynamics and celestial mechanics, developing a specialized computer algebra system for high-order Lie transformations. …

32: Richard B. Paris

… ►His books are Asymptotics of High Order Differential Equations (with A. …

33: 10.1 Special Notation

… ► ►►►

$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
…
$\nu$	real or complex parameter (the order).
…

… ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

34: 14.10 Recurrence Relations and Derivatives

… ►

14.10.1 $${𝖯}_{\nu }^{\mu +2}\left(x\right)+2(\mu +1)x{\left(1-x^2\right)}^{-1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)+(\nu -\mu )(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.2 $${\left(1-x^2\right)}^{1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.3 $$(\nu -\mu +2){𝖯}_{\nu +2}^{\mu }\left(x\right)-(2\nu +3)x{𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.3 (modified)

Referenced by:

§14.10, §14.14, §14.21(iii)

Permalink:

http://dlmf.nist.gov/14.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.4 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\mu -\nu -1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +1)x{𝖯}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.5(i)

Permalink:

http://dlmf.nist.gov/14.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

►

14.10.5 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){𝖯}_{\nu -1}^{\mu }\left(x\right)-\nu x{𝖯}_{\nu }^{\mu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.4 (modified)

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

…

35: 10.74 Methods of Computation

… ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). … ►In the case of $J_n\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). … ►

§10.74(viii) Functions of Imaginary Order

►For the computation of the functions ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).

36: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

►With real critical points (36.4.1) ordered so that … ►

36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

… ►

36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

37: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

►

19.27.3 $$R_F(x,y,z)=R_F(0,y,z)-\frac{1}{\sqrt{h}}\left(\sqrt{\frac{x}{h}}+O\left(\frac{x}{h}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.5 $$R_G(x,y,z)=R_G(0,y,z)+\sqrt{x}O\left(\sqrt{x/h}\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

…

38: 11.10 Anger–Weber Functions

… ►

11.10.5 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=f(\nu ,z),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/11.10.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

11.10.6 $$f(\nu ,z)=\frac{(z-\nu )}{\pi z^2}\sin \left(\pi \nu \right),$$ $w={𝐉}_{\nu }\left(z\right)$,

ⓘ

Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E6

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

… ►

11.10.34 $$2{𝐉}_{\nu }^{\prime }\left(z\right)={𝐉}_{\nu -1}\left(z\right)-{𝐉}_{\nu +1}\left(z\right),$$

ⓘ

Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E34

Encodings:

pMML, png, TeX

See also:

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

►

11.10.35 $$2{𝐄}_{\nu }^{\prime }\left(z\right)={𝐄}_{\nu -1}\left(z\right)-{𝐄}_{\nu +1}\left(z\right),$$

ⓘ

Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E35

Encodings:

pMML, png, TeX

See also:

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

…

39: Bibliography T

… ►

Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

… ►

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

… ►

Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

►

Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

►

Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.

ⓘ

External Links:

MathReview Entry

Referenced by:

(1.18.59), §1.18(ix), §1.18(v), §1.18(vi), §1.18(x), (10.22.78), §10.22(v), §18.39(ii).

Encodings:

BibTeX

…

40: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ► ►

§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).