modulus and phase functions

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21: 33.2 Definitions and Basic Properties

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33.2.7

{H^{\pm}_{\ell}}\left(\eta,\rho\right)=(\mp\mathrm{i})^{\ell}e^{(\pi\eta/2)\pm% \mathrm{i}{\sigma_{\ell}}\left(\eta\right)}W_{\mp\mathrm{i}\eta,\ell+\frac{1}{% 2}}\left(\mp 2\mathrm{i}\rho\right),

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Defines:: ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions
Symbols:: ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter
Referenced by:: §13.18(vi), §33.23(vi)
Permalink:: http://dlmf.nist.gov/33.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §33.2(iii), §33.2 and Ch.33

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22: 10.21 Zeros

… ►

\phi_{\nu}\left({y^{\prime}_{\nu,m}}\right)=m\pi,

m=1,2,\dotsc

… ►Let

\mathscr{C}_{\nu}\left(x\right)

\rho_{\nu}(t)

, and

\sigma_{\nu}(t)

be defined as in §10.21(ii) and

M\left(x\right)

\theta\left(x\right)

N\left(x\right)

, and

\phi\left(x\right)

denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8. … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. …

23: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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33.11.1

{H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},

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Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $k$ : nonnegative integer, $\ell$ : 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

►where

{\theta_{\ell}}\left(\eta,\rho\right)

is defined by (33.2.9), and

a

and

b

are defined by (33.8.3). … ►

33.11.4

{H^{\pm}_{\ell}}\left(\eta,\rho\right)=e^{\pm\mathrm{i}{\theta_{\ell}}}(f(\eta% ,\rho)\pm\mathrm{i}g(\eta,\rho)),

ⓘ

Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : 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

… ►Here

f_{0}=1

g_{0}=0

\widehat{f}_{0}=0

\widehat{g}_{0}=1-(\eta/\rho)

, and for

k=0,1,2,\dots

, …

24: 4.45 Methods of Computation

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Logarithms

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Exponentials

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Trigonometric Functions

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Inverse Trigonometric Functions

… ►See §1.9(i) for the precise relationship of

\operatorname{ph}z

to the arctangent function. …

25: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►Then as

a\to\infty

in the sector

|\operatorname{ph}a|\leq\pi-\delta(<\pi)

, …in each case uniformly with respect to

\lambda

in the sector

|\operatorname{ph}\lambda|\leq 2\pi-\delta

(

<2\pi

). … ►for

z\to\infty

\left|\operatorname{ph}z\right|<\frac{1}{2}\pi

, with

\Re\left(z-a\right)\leq 0

for

P\left(a,z\right)

and

\Re\left(z-a\right)\geq 0

for

Q\left(a,z\right)

. … ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function

\operatorname{erfc}

see Paris (2002b) and Dunster (1996a). ►

Inverse Function

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26: 10.75 Tables

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•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_{0}\left(x\right)$ to 15D, $J_{1}\left(x\right)$ , $J_{2}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ to 10D, $Y_{2}\left(x\right)$ to 8D, $x=0(.1)17.5$ ; $Y_{n}\left(x\right)-(2/\pi)J_{n}\left(x\right)\ln x$ , $n=0,1$ , $x=0(.1)2$ , 8D; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)20$ , 5D or 5S; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 10S; modulus and phase functions $\sqrt{x}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-x$ , $n=0,1,2$ , $\ifrac{1}{x}=0(.01)0.1$ , 8D.

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•

Young and Kirk (1964) tabulates $\operatorname{ber}_{n}x$ , $\operatorname{bei}_{n}x$ , $\operatorname{ker}_{n}x$ , $\operatorname{kei}_{n}x$ , $n=0,1$ , $x=0(.1)10$ , 15D; $\operatorname{ber}_{n}x$ , $\operatorname{bei}_{n}x$ , $\operatorname{ker}_{n}x$ , $\operatorname{kei}_{n}x$ , modulus and phase functions $M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)$ , $N_{n}\left(x\right)$ , $\phi_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.01)2.5$ , 8S, and $n=0(1)10$ , $x=0(.1)10$ , 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$ . (Concerning the phase functions see §10.68(iv).)

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Abramowitz and Stegun (1964, Chapter 9) tabulates $\operatorname{ber}_{n}x$ , $\operatorname{bei}_{n}x$ , $\operatorname{ker}_{n}x$ , $\operatorname{kei}_{n}x$ , $n=0,1$ , $x=0(.1)5$ , 9–10D; $x^{n}(\operatorname{ker}_{n}x+(\operatorname{ber}_{n}x)(\ln x))$ , $x^{n}(\operatorname{kei}_{n}x+(\operatorname{bei}_{n}x)(\ln x))$ , $n=0,1$ , $x=0(.1)1$ , 9D; modulus and phase functions $M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)$ , $N_{n}\left(x\right)$ , $\phi_{n}\left(x\right)$ , $n=0,1$ , $x=0(.2)7$ , 6D; $\sqrt{x}e^{-x/\sqrt{2}}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-(x/\sqrt{2})$ , $\sqrt{x}e^{x/\sqrt{2}}N_{n}\left(x\right)$ , $\phi_{n}\left(x\right)+(x/\sqrt{2})$ , $n=0,1$ , $1/x=0(.01)0.15$ , 5D.

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27: 9.3 Graphics

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§9.3(i) Real Variable

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28: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Confluent Hypergeometric Functions

… ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors

\tfrac{1}{2}\pi<\left|\operatorname{ph}z\right|<\pi

one has to use the analytic continuation formula (13.2.12). ►

Parabolic Cylinder Functions

… ►

Probability Functions

…

29: 36.12 Uniform Approximation of Integrals

… ►As

\mathbf{y}

varies as many as

K+1

(real or complex) critical points of the smooth phase function

f

can coalesce in clusters of two or more. The function

g

has a smooth amplitude. Also,

f

is real analytic, and

\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}>0

for all

\mathbf{y}

such that all

K+1

critical points coincide. If

\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}<0

, then we may evaluate the complex conjugate of

I

for real values of

\mathbf{y}

and

g

, and obtain

I

by conjugation and analytic continuation. … ►Also,

\Delta^{1/4}/\sqrt{f_{+}^{\prime\prime}}

and

\Delta^{1/4}/\sqrt{-f_{-}^{\prime\prime}}

are chosen to be positive real when

y

is such that both critical points are real, and by analytic continuation otherwise. …

30: 15.8 Transformations of Variable

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15.8.14

F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1% }{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),

|\operatorname{ph}\left(1-z\right)|<\pi

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

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