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11: Bibliography B

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Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.

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External Links:

ISBN 0-387-96794-X, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§15.7.

Encodings:

BibTeX

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Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 0-387-97503-9, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

Encodings:

BibTeX

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Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

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External Links:

Document

Referenced by:

§36.14(i).

Encodings:

BibTeX

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Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

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External Links:

ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

Encodings:

BibTeX

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12: 10.45 Functions of Imaginary Order

… ►With $z=x$, and $\nu$ replaced by $\mathrm{i}\nu$, the modified Bessel’s equation (10.25.1) becomes … ►

10.45.8 $${\tilde{K}}_0\left(x\right)=K_0\left(x\right)=-\ln \left(\frac{1}{2}x\right)-\gamma +O\left(x^2\ln x\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $\ln z$: principal branch of logarithm function and $x$: real variable

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

►where $\gamma$ again denotes Euler’s constant (§5.2(ii)). … ►For properties of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …

13: 10.24 Functions of Imaginary Order

… ►With $z=x$ and $\nu$ replaced by $\mathrm{i}\nu$, Bessel’s equation (10.2.1) becomes … ►

10.24.9 $${\tilde{Y}}_0\left(x\right)=Y_0\left(x\right)=\frac{2}{\pi }\left(\ln \left(\frac{1}{2}x\right)+\gamma \right)+O\left(x^2\ln x\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

►where $\gamma$ denotes Euler’s constant §5.2(ii). … ►For mathematical properties and applications of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …

14: 14.23 Values on the Cut

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14.23.2 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}\left({𝖰}_{\nu }^{\mu }\left(x\right)\mp \frac{1}{2}\pi \mathrm{i}{𝖯}_{\nu }^{\mu }\left(x\right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.3 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{\pi }^{3/2}{\left(1-x^2\right)}^{\mu /2}}{2^{\nu +1}}\left(\frac{x𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\nu +\frac{1}{2}\mu +1;\frac{3}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}\mp \mathrm{i}\frac{𝐅(\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2};\frac{1}{2};x^2)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +1\right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.5 $${𝖰}_{\nu }^{\mu }\left(x\right)=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.4 (modified)

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.6 $${𝖰}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)\pm \frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23, §14.23

Permalink:

http://dlmf.nist.gov/14.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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15: 15.12 Asymptotic Approximations

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15.12.2 $$F(a,b;c;z)=\sum _{s=0}^{m-1}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s+O\left(c^{-m}\right),$$ $|c|\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

Referenced by:

§15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

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15.12.3 $$F(\genfrac{}{}{0pt}{}{a,b}{c+\lambda };z)\sim \frac{\mathrm{\Gamma }\left(c+\lambda \right)}{\mathrm{\Gamma }\left(c-b+\lambda \right)}\sum _{s=0}^{\mathrm{\infty }}{q}_s(z){\left(b\right)}_s{\lambda }^{-s-b},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $q_s(z)$: coefficients

Referenced by:

§15.12(iii), §15.12(iii), §15.19(iv)

Permalink:

http://dlmf.nist.gov/15.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

►where $q_0(z)=1$ and $q_s(z)$, $s=1,2,\mathrm{\dots }$, are defined by the generating function ►

15.12.4 $${\left(\frac{{\mathrm{e}}^t-1}{t}\right)}^{b-1}{\mathrm{e}}^{t(1-c)}{\left(1-z+z{\mathrm{e}}^{-t}\right)}^{-a}=\sum _{s=0}^{\mathrm{\infty }}{q}_s(z)t^s.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $q_s(z)$: coefficients

Permalink:

http://dlmf.nist.gov/15.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1). …

16: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►For further information, including extensions to complex values of $x$ and $p$, see Temme (1994b, §4) and Dunster (1996b, 1997).

17: Errata

… ►We now include Markov’s Theorem. In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. … ►

The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

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10.23.11 $$a_k=\frac{1}{2\pi i}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported by Mark Dunster on 2021-03-22

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The original constraint, $\mathrm{\Re }s>0$, was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right)-{(s-1)}^{-1}$ is an entire function.

Suggested by John Harper.

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