俄罗斯环境科学学位证书【somewhat微aptao168】rho

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1: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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

33.11.7 $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

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Symbols:

$\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function, $g(\eta ,\rho )$: function, $\widehat{f}(\eta ,\rho )$: function and $\widehat{g}(\eta ,\rho )$: function

A&S Ref:

14.5.8

Referenced by:

§33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7)

Permalink:

http://dlmf.nist.gov/33.11.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in this equation 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 $k=0,1,2,\mathrm{\dots }$, …

2: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

►As $\rho \to 0$ with $\eta$ fixed, … ►

§33.5(ii) $\eta =0$

… ►

§33.5(iv) Large $\mathrm{\ell }$

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3: 33.2 Definitions and Basic Properties

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

… ► … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . … ►

§33.2(iv) Wronskians and Cross-Product

►With arguments $\eta ,\rho$ suppressed, …

4: 33.3 Graphics

§33.3 Graphics

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§33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$

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Figure 33.3.6: $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $M_{\mathrm{\ell }}(\eta ,\rho )$ with $\mathrm{\ell }=5$, $\eta =0$. … Magnify

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§33.3(ii) Surfaces of the Coulomb Radial Functions $F_0(\eta ,\rho )$ and $G_0(\eta ,\rho )$

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Figure 33.3.8: $G_0(\eta ,\rho )$, $-2\le \eta \le 2$, . Magnify 3D Help

5: 33.24 Tables

§33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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6: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

… ►

§33.10(i) Large $\rho$

►As $\rho \to \mathrm{\infty }$ with $\eta$ fixed, …where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9). … ►As $\eta \to \mathrm{\infty }$ with $\eta \rho$ fixed, … ►As $\eta \to -\mathrm{\infty }$ with $\eta \rho$ fixed, …

7: Software Index

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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8: 18.31 Bernstein–Szegő Polynomials

… ►Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell }$ and positive when $-1\le x\le 1$. The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. In consequence, $p_n(\cos \theta )$ can be given explicitly in terms of $\rho (\cos \theta )$ and sines and cosines, provided that in the first case, in the second case, and in the third case. …

9: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►

§33.23(vii) WKBJ Approximations

►WKBJ approximations (§2.7(iii)) for $\rho >{\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq.  (12) $(\rho -c)/c$ should be $(\rho -c)/\rho$). …

10: 33.7 Integral Representations

§33.7 Integral Representations

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33.7.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=\frac{{\rho }^{\mathrm{\ell }+1}{2}^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}\rho -(\pi \eta /2)}}{|\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)|}{\int }_0^1{\mathrm{e}}^{-2\mathrm{i}\rho t}{t}^{\mathrm{\ell }+\mathrm{i}\eta }{(1-t)}^{\mathrm{\ell }-\mathrm{i}\eta }dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§33.7, §33.7

Permalink:

http://dlmf.nist.gov/33.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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33.7.2 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{{\mathrm{e}}^{-\mathrm{i}\rho }{\rho }^{-\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\mathrm{\ell }-\mathrm{i}\eta }{(t+2\mathrm{i}\rho )}^{\mathrm{\ell }+\mathrm{i}\eta }dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.1 (modified)

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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33.7.3 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{-\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}\left(\frac{\exp \left(-\mathrm{i}(\rho \tanh t-2\eta t)\right)}{{(\cosh t)}^{2\mathrm{\ell }+2}}+\mathrm{i}{(1+t^2)}^{\mathrm{\ell }}\exp \left(-\rho t+2\eta \arctan t\right)\right)dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\arctan z$: arctangent function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.3

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

►

33.7.4 $$H_{\mathrm{\ell }}^{+}(\eta ,\rho )=\frac{\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_{-1}^{-\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}\rho t}{(1-t)}^{\mathrm{\ell }-\mathrm{i}\eta }{(1+t)}^{\mathrm{\ell }+\mathrm{i}\eta }dt.$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.2

Referenced by:

§33.7, §33.7

Permalink:

http://dlmf.nist.gov/33.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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