俄罗斯环境科学学位证书【somewhat微aptao168】rho

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11: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►The function $\varphi (\rho ,\beta ;z)$ is defined by ►

10.46.1 $$\varphi (\rho ,\beta ;z)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{k!\mathrm{\Gamma }\left(\rho k+\beta \right)},$$ $\rho >-1$.

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Defines:

$\varphi (\rho ,\beta ;z)$: generalized Bessel function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.46.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

… ►For asymptotic expansions of $\varphi (\rho ,\beta ;z)$ as $z\to \mathrm{\infty }$ in various sectors of the complex $z$-plane for fixed real values of $\rho$ and fixed real or complex values of $\beta$, see Wright (1935) when $\rho >0$, and Wright (1940b) when . For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when $\rho >0$, and Wong and Zhao (1999a) when . ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: …

12: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

►

33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

►

33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

… ►The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. Corresponding expansions for $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

13: 33.8 Continued Fractions

§33.8 Continued Fractions

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

33.8.2 $$\frac{H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }}{H_{\mathrm{\ell }}^{\pm }}=c\pm \frac{\mathrm{i}}{\rho }\frac{ab}{2(\rho -\eta \pm \mathrm{i})+}\frac{(a+1)(b+1)}{2(\rho -\eta \pm 2\mathrm{i})+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient, $b$: coefficient and $c$: coefficient

Referenced by:

§33.8

Permalink:

http://dlmf.nist.gov/33.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

… ►

$c=\pm \mathrm{i}(1-(\eta /\rho )).$

►The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho \ne 0$. …

14: 36.13 Kelvin’s Ship-Wave Pattern

… ►The integral is of the form of the real part of (36.12.1) with $y=\varphi$, $u=\theta$, $g=1$, $k=\rho$, and … ►When $\rho >1$, that is, everywhere except close to the ship, the integrand oscillates rapidly. … ►The disturbance $z(\rho ,\varphi )$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that ${\theta }_{\pm }(\varphi )$ are real for and complex for $|\varphi |>{\varphi }_c$. … ►

36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

… ►

►►

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

…

15: 33.9 Expansions in Series of Bessel Functions

… ►

§33.9(i) Spherical Bessel Functions

… ►The series (33.9.1) converges for all finite values of $\eta$ and $\rho$. ►

§33.9(ii) Bessel Functions and Modified Bessel Functions

… ►With $t=2\left|\eta \right|\rho$, … ►Next, as $\eta \to +\mathrm{\infty }$ with $\rho$ ($>0$) fixed, …

16: 33.13 Complex Variable and Parameters

§33.13 Complex Variable and Parameters

►The functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ may be extended to noninteger values of $\mathrm{\ell }$ by generalizing $(2\mathrm{\ell }+1)!=\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U(a,b,z)$ expanded via (13.2.42). ►These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\mathrm{\ell }$. …

17: 33.4 Recurrence Relations and Derivatives

§33.4 Recurrence Relations and Derivatives

… ►

$S_{\mathrm{\ell }}={\displaystyle \frac{\mathrm{\ell }}{\rho }}+{\displaystyle \frac{\eta }{\mathrm{\ell }}},$

… ►Then, with $X_{\mathrm{\ell }}$ denoting any of $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, or $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$, …

18: 2.9 Difference Equations

… ►where ${\rho }_1,{\rho }_2$ are the roots of the characteristic equation … ►The construction fails if ${\rho }_1={\rho }_2$, that is, when $f_0^2=4g_0$. … ►When $|{\rho }_2|=|{\rho }_1|$ and $\mathrm{\Re }{\alpha }_2=\mathrm{\Re }{\alpha }_1$ neither solution is dominant and both are unique. … ►When the roots of (2.9.5) are equal we denote them both by $\rho$. Assume first $2g_1\ne f_0{f}_1$. …

19: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

►

§33.12(i) Transition Region

… ►

§33.12(ii) Uniform Expansions

►With the substitution $\rho =2\eta z$, Equation (33.2.1) becomes … ►

20: 33.25 Approximations

§33.25 Approximations

…