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11: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

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33.11.4 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function and $g(\eta ,\rho )$: function

Permalink:

http://dlmf.nist.gov/33.11.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in these equations were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

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12: 14.23 Values on the Cut

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14.23.1 $$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{𝖯}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: 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.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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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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14.23.4 $${𝖯}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: 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:

8.3.2 (corrected)

Referenced by:

§14.23, §14.23, §14.3(iv)

Permalink:

http://dlmf.nist.gov/14.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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

ⓘ

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

►If cuts are introduced along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$, then (14.23.4) and (14.23.6) could be used to extend the definitions of ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ to complex $x$. …

13: 12.19 Tables

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Abramowitz and Stegun (1964, Chapter 19) includes $U(a,x)$ and $V(a,x)$ for $\pm a=0(.1)1(.5)5$, $x=0(.1)5$, 5S; $W(a,\pm x)$ for $\pm a=0(.1)1(1)5$, $x=0(.1)5$, 4-5D or 4-5S.

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Kireyeva and Karpov (1961) includes $D_p\left(x(1+\mathrm{i})\right)$ for $\pm x=0(.1)5$, $p=0(.1)2$, and $\pm x=5(.01)10$, $p=0(.5)2$, 7D.

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Karpov and Čistova (1964) includes $D_p\left(x\right)$ for $p=-2(.1)0$, $\pm x=0(.01)5$; $p=-2(.05)0$, $\pm x=5(.01)10$, 6D.

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Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

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14: 10.27 Connection Formulas

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10.27.5 $$K_n\left(z\right)=\frac{{(-1)}^{n-1}}{2}\left({\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n}\right),$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.6 $$I_{\nu }\left(z\right)={\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{J}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.3 (modified)

Referenced by:

§10.26(ii), §10.27, §10.27, §10.31, §10.33, §10.35, §10.39, §10.41(i), §10.42, §10.43(ii), §10.43(iv), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), §10.67(i), §11.4(i)

Permalink:

http://dlmf.nist.gov/10.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.7 $$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.11 $$Y_{\nu }\left(z\right)={\mathrm{e}}^{\pm (\nu +1)\pi \mathrm{i}/2}{I}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right)-(2/\pi ){\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right),$$ $-\frac{1}{2}\pi \le \pm \mathrm{ph}z\le \pi$.

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.5 (modified)

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/10.27.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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15: 14.9 Connection Formulas

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§14.9(i) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖯}_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝖰}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(x\right)$

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§14.9(ii) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$

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§14.9(iii) Connections Between $P_{\nu }^{\pm \mu }\left(x\right)$, $P_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝑸}_{\nu }^{\pm \mu }\left(x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(x\right)$

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16: 15.19 Methods of Computation

… ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . … ►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …

17: 4.23 Inverse Trigonometric Functions

… ►In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points $t=\pm 1$. …In (4.23.3) the integration path may not intersect $\pm \mathrm{i}$. …$\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►For the corresponding results for $\mathrm{arccsc}z$, $\mathrm{arcsec}z$, and $\mathrm{arccot}z$, use (4.23.7)–(4.23.9). … ►where $z=x+\mathrm{i}y$ and $\pm z\notin (1,\mathrm{\infty })$ in (4.23.34) and (4.23.35), and in (4.23.36). …

18: 26.18 Counting Techniques

… ►Then the number of elements in the set $S\setminus (A_1\cup A_2\cup \mathrm{\cdots }\cup A_n)$ is ►

26.18.1

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Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

… ►For further examples in the use of generating functions, see Stanley (1997, 1999) and Wilf (1994). …

19: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

20: 4.16 Elementary Properties

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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

► ►►►►►►

$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
$\sin x$	$-\sin \theta$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$
$\cos x$	$\cos \theta$	$\mp \sin \theta$	$-\cos \theta$	$\pm \sin \theta$	$\cos \theta$
$\tan x$	$-\tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$
…
$\cot x$	$-\cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$

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