relation to modulus and phase

SearchAdvancedHelp

(0.008 seconds)

1—10 of 53 matching pages

1: 9.9 Zeros

… ►

§9.9(ii) Relation to Modulus and Phase

…

2: 9.8 Modulus and Phase

… ►(These definitions of $\theta \left(x\right)$ and $\varphi \left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) …

3: 10.21 Zeros

… ►

${\varphi }_{\nu }\left(y_{\nu ,m}^{\prime }\right)=m\pi ,$ $m=1,2,\mathrm{\dots }$.

…

4: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

… ►

Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$-plane that exclude $z=0$ and are valid for .

… ►

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

►

Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_1\left(z\right)$ and $z^{-1}{\int }_0^z{t}^{-1}(1-{\mathrm{e}}^{-t})dt$ for complex $z$ with $\left|\mathrm{ph}z\right|\le \pi$.

►

Verbeeck (1970) gives polynomial and rational approximations for $E_p\left(x\right)=({\mathrm{e}}^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.

5: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

►

§10.68(i) Definitions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►

§10.68(ii) Basic Properties

… ►10)), and ${lim}_{x\to \mathrm{\infty }}({\varphi }_1\left(x\right)+(x/\sqrt{2}))=-\frac{5}{8}\pi$ (Eqs. …

6: 33.11 Asymptotic Expansions for Large $\rho$

… ►

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

ⓘ

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

►where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3). … ►

$F_{\mathrm{\ell }}(\eta ,\rho )=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

… ►

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

ⓘ

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

… ►Here $f_0=1$, $g_0=0$, ${\widehat{f}}_0=0$, ${\widehat{g}}_0=1-(\eta /\rho )$, and for $k=0,1,2,\mathrm{\dots }$, …

7: 33.2 Definitions and Basic Properties

… ►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by … ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by …the branch of the phase in (33.2.10) being zero when $\eta =0$ and continuous elsewhere. ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift. …

8: 9.7 Asymptotic Expansions

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►As $z\to \mathrm{\infty }$ the following asymptotic expansions are valid uniformly in the stated sectors. … ►where $\xi =\frac{2}{3}{x}^{3/2}$. … ►Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms. … ►And as $z\to \mathrm{\infty }$ with $m$ fixed …

9: 15.8 Transformations of Variable

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►The hypergeometric functions that correspond to Groups 1 and 2 have $z$ as variable. The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of $z$ as variable. … ►With $\zeta ={\mathrm{e}}^{2\pi \mathrm{i}/3}(1-z)/\left(z-{\mathrm{e}}^{4\pi \mathrm{i}/3}\right)$ … ►provided that $z$ lies in the intersection of the open disks , or equivalently, . …

10: 8.11 Asymptotic Approximations and Expansions

… ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. … ►

§8.11(iii) Large $a$, Fixed $z/a$

… ►

§8.11(iv) Large $a$, Bounded $(x-a)/{(2a)}^{\frac{1}{2}}$

… ►in both cases uniformly with respect to bounded real values of $y$. …For related expansions involving Hermite polynomials see Pagurova (1965). …