毕业证跟学位证书的区别〖办证V信ATV1819〗whitm

SearchAdvancedHelp

(0.002 seconds)

1—10 of 26 matching pages

1: 13.25 Products

… ►

13.25.1 $$M_{\kappa ,\mu }\left(z\right)M_{\kappa ,-\mu -1}\left(z\right)+\frac{(\frac{1}{2}+\mu +\kappa )(\frac{1}{2}+\mu -\kappa )}{4\mu (1+\mu ){(1+2\mu )}^2}{M}_{\kappa ,\mu +1}\left(z\right)M_{\kappa ,-\mu }\left(z\right)=1.$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.25

Permalink:

http://dlmf.nist.gov/13.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.25 and Ch.13

►For integral representations, integrals, and series containing products of $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).

2: 13.15 Recurrence Relations and Derivatives

… ►

13.15.1 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -1,\mu }\left(z\right)+(z-2\kappa )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})M_{\kappa +1,\mu }\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.29

Permalink:

http://dlmf.nist.gov/13.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.2 $$2\mu (1+2\mu )\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-(z+2\mu )(1+2\mu )M_{\kappa ,\mu }\left(z\right)+(\kappa +\mu +\frac{1}{2})\sqrt{z}{M}_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.15(i)

Permalink:

http://dlmf.nist.gov/13.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.3 $$(\kappa -\mu -\frac{1}{2})M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)+(1+2\mu )\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)-(\kappa +\mu +\frac{1}{2})M_{\kappa +\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.4 $$2\mu M_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-2\mu M_{\kappa +\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.28

Permalink:

http://dlmf.nist.gov/13.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

►

13.15.5 $$2\mu (1+2\mu )M_{\kappa ,\mu }\left(z\right)-2\mu (1+2\mu )\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)-(\kappa -\mu -\frac{1}{2})\sqrt{z}{M}_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)=0,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.15(i), §13.15 and Ch.13

…

3: 13.18 Relations to Other Functions

… ►

13.18.1 $$M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z,$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

►

13.18.2 $$M_{\kappa ,\kappa -\frac{1}{2}}\left(z\right)=W_{\kappa ,\kappa -\frac{1}{2}}\left(z\right)=W_{\kappa ,-\kappa +\frac{1}{2}}\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa },$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

►

13.18.3 $$M_{\kappa ,-\kappa -\frac{1}{2}}\left(z\right)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }.$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

… ►

13.18.4 $$M_{\mu -\frac{1}{2},\mu }\left(z\right)=2\mu {\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{2}-\mu }\gamma (2\mu ,z),$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/13.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

… ►

13.18.6 $$M_{-\frac{1}{4},\frac{1}{4}}\left(z^2\right)=\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erf}\left(z\right),$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

…

4: 13.14 Definitions and Basic Properties

… ►except that $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$. … ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. …

5: 13.26 Addition and Multiplication Theorems

… ►

§13.26(i) Addition Theorems for $M_{\kappa ,\mu }\left(z\right)$

►The function $M_{\kappa ,\mu }\left(x+y\right)$ has the following expansions: … ►

13.26.2 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

… ►

§13.26(iii) Multiplication Theorems for $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$

►To obtain similar expansions for $M_{\kappa ,\mu }\left(xy\right)$ and $W_{\kappa ,\mu }\left(xy\right)$, replace $y$ in the previous two subsections by $(y-1)x$.

6: 13.32 Software

…

7: 13.24 Series

… ►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). … ►

13.24.1 $$M_{\kappa ,\mu }\left(z\right)=\mathrm{\Gamma }\left(\kappa +\mu \right)2^{2\kappa +2\mu }{z}^{\frac{1}{2}-\kappa }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2\kappa +2\mu \right)}_s{\left(2\kappa \right)}_s}{{\left(1+2\mu \right)}_{s}s!}\left(\kappa +\mu +s\right)I_{\kappa +\mu +s}\left(\frac{1}{2}z\right),$$ $2\mu ,\kappa +\mu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.11.2), §13.24(ii)

Permalink:

http://dlmf.nist.gov/13.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

… ►

13.24.2 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=2^{2\mu }{z}^{\mu +\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(2\sqrt{\kappa z}\right)}^{-2\mu -s}{J}_{2\mu +s}\left(2\sqrt{\kappa z}\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

…

8: 13.22 Zeros

… ►From (13.14.2) and (13.14.3) $M_{\kappa ,\mu }\left(z\right)$ has the same zeros as $M(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ and $W_{\kappa ,\mu }\left(z\right)$ has the same zeros as $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$, hence the results given in §13.9 can be adopted. … ►For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by …

9: 8.5 Confluent Hypergeometric Representations

… ►For the confluent hypergeometric functions $M$, $𝐌$, $U$, and the Whittaker functions $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$, see §§13.2(i) and 13.14(i). … ►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

…

10: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions $M(a,b,z)$ and $U(a,b,z)$, Olver’s function $𝐌(a,b,z)$, and the Whittaker functions $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$. … ►Other notations are: ${}_1{}^{}F_1^{}(a;b;z)$ (§16.2(i)) and $\mathrm{\Phi }(a;b;z)$ (Humbert (1920)) for $M(a,b,z)$; $\mathrm{\Psi }(a;b;z)$ (Erdélyi et al. (1953a, §6.5)) for $U(a,b,z)$; $V(b-a,b,z)$ (Olver (1997b, p. 256)) for ${\mathrm{e}}^{z}U(a,b,-z)$; $\mathrm{\Gamma }\left(1+2\mu \right){ℳ}_{\kappa ,\mu }$ (Buchholz (1969, p. 12)) for $M_{\kappa ,\mu }\left(z\right)$. …