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31: 5.20 Physical Applications

§5.20 Physical Applications

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Rutherford Scattering

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5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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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, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

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Annotations for §5.20, §5.20 and Ch.5

… ►Carlitz (1972) describes the partition function of dense hadronic matter in terms of a gamma function.

32: 5.14 Multidimensional Integrals

§5.14 Multidimensional Integrals

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5.14.1 $${\int }_{V_n}{t}_1^{z_1-1}{t}_2^{z_2-1}\mathrm{\cdots }{t}_n^{z_n-1}d{t}_{1}d{t}_2\mathrm{\cdots }d{t}_n=\frac{\mathrm{\Gamma }\left(z_1\right)\mathrm{\Gamma }\left(z_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(z_n\right)}{\mathrm{\Gamma }\left(1+z_1+z_2+\mathrm{\cdots }+z_n\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $z$: complex variable and $V_n$: simplex

Referenced by:

§5.19(iii)

Permalink:

http://dlmf.nist.gov/5.14.E1

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… ►

Selberg-type Integrals

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Dyson’s Integral

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5.14.7 $\mathrm{\Re }b>-1/n$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $b$: real or complex variable

Referenced by:

§5.14, §5.20

Permalink:

http://dlmf.nist.gov/5.14.E7

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33: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

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§5.11(ii) Error Bounds and Exponential Improvement

… ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). ►

§5.11(iii) Ratios

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34: 8.10 Inequalities

§8.10 Inequalities

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8.10.1 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)\le 1,$$ $x>0$, ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable and $a$: parameter

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E1

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8.10.2 $$\gamma (a,x)\ge \frac{x^{a-1}}{a}(1-{\mathrm{e}}^{-x}),$$ $x>0$, .

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $x$: real variable and $a$: parameter

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E2

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Padé Approximants

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8.10.13 $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $n$: nonnegative integer

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E13

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35: 35.3 Multivariate Gamma and Beta Functions

§35.3 Multivariate Gamma and Beta Functions

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§35.3(i) Definitions

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§35.3(ii) Properties

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35.3.6 $${\mathrm{\Gamma }}_m\left(a,\mathrm{\dots },a\right)={\mathrm{\Gamma }}_m\left(a\right).$$

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Symbols:

${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E6

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Annotations for §35.3(ii), §35.3 and Ch.35

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35.3.7 $${\mathrm{B}}_m(a,b)=\frac{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(b\right)}{{\mathrm{\Gamma }}_m\left(a+b\right)}.$$

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Symbols:

${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E7

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36: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.3 $$P(a,z)=\frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{a/2}\right)-S(a,\eta ),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E3

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$c_5(0)=-\frac{\mathrm{27\hspace{0.33em}45493}}{\mathrm{81517\hspace{0.33em}36320}}.$

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). ►

Inverse Function

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37: 5.9 Integral Representations

§5.9 Integral Representations

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§5.9(i) Gamma Function

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Hankel’s Loop Integral

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Binet’s Formula

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38: 5.4 Special Values and Extrema

§5.4 Special Values and Extrema

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§5.4(i) Gamma Function

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5.4.7 $$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.11 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073005; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E7

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5.4.11 $${\mathrm{\Gamma }}^{\prime }\left(1\right)=-\gamma .$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\gamma$: Euler’s constant

A&S Ref:

6.3.2

Referenced by:

§5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E11

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§5.4(iii) Extrema

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39: 8.1 Special Notation

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$x$	real variable.
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$\mathrm{\Gamma }\left(z\right)$	gamma function (§5.2(i)).
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►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

40: Ranjan Roy

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5 Gamma Function

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