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11: 32.2 Differential Equations

… ►with $\alpha$, $\beta$, $\gamma$, and $\delta$ arbitrary constants. … ►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. … ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. … ►If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. …Lastly, if $\delta =0$ and $\beta \gamma \ne 0$, then set $\beta =-1$ and $\gamma =1$, without loss of generality. …

12: 8.8 Recurrence Relations and Derivatives

… ►

8.8.1 $$\gamma (a+1,z)=a\gamma (a,z)-z^a{\mathrm{e}}^{-z},$$

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Symbols:

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

A&S Ref:

6.5.22

Permalink:

http://dlmf.nist.gov/8.8.E1

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Annotations for §8.8 and Ch.8

… ►If $w(a,z)=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then … ►

8.8.4 $$z{\gamma }^{\ast }(a+1,z)={\gamma }^{\ast }(a,z)-\frac{{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.8.E4

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Annotations for §8.8 and Ch.8

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8.8.8 $$\gamma (a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\gamma (a-n,z)-z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E8

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Annotations for §8.8 and Ch.8

… ►

8.8.10 $$\mathrm{\Gamma }(a,z)=\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-n\right)}\mathrm{\Gamma }(a-n,z)+z^{a-1}{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(a-k\right)}{z}^{-k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E10

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Annotations for §8.8 and Ch.8

…

13: 5.8 Infinite Products

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5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

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

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

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Annotations for §5.8 and Ch.5

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5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

Encodings:

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Annotations for §5.8 and Ch.5

►

5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

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Annotations for §5.8 and Ch.5

… ►

5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

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Annotations for §5.8 and Ch.5

…

14: 15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. …Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). … ►where $\kappa$, $\lambda$, $\mu$, $\nu$ are real or complex constants such that $\kappa \nu -\lambda \mu =1$. These constants can be chosen to map any two sets of three distinct points $\{\alpha ,\beta ,\gamma \}$ and $\{\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\}$ onto each other. …

15: 30.4 Functions of the First Kind

… ►The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. …the sign of ${\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)$ being ${(-1)}^{(n+m)/2}$ when $n-m$ is even, and the sign of ${d{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)/dx|}_{x=0}$ being ${(-1)}^{(n+m-1)/2}$ when $n-m$ is odd. ►When ${\gamma }^2>0$ ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the prolate angular spheroidal wave function, and when ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the oblate angular spheroidal wave function. If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$: … ► ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ has exactly $n-m$ zeros in the interval . …

16: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. ►The function ${\gamma }^{\ast }(a,z)$ is entire in $z$ and $a$. … ►If $w=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then … ►If $w={\mathrm{e}}^z{z}^{1-a}\mathrm{\Gamma }(a,z)$, then …

17: 8.1 Special Notation

… ► ►►►►►

$x$	real variable.
…
$\delta$	arbitrary small positive constant.
$\mathrm{\Gamma }\left(z\right)$	gamma function (§5.2(i)).
$\psi \left(z\right)$	${\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right)$.

… ►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).

18: 8.7 Series Expansions

… ►

8.7.1 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}=\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(a+k)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.29

Referenced by:

§8.11(ii), §8.14, §8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.7.E1

Encodings:

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Annotations for §8.7 and Ch.8

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8.7.2 $$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.30

Permalink:

http://dlmf.nist.gov/8.7.E2

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Annotations for §8.7 and Ch.8

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8.7.3 $$\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{a+k}}{k!(a+k)}=\mathrm{\Gamma }\left(a\right)\left(1-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.14, §8.19(iv), §8.4, §8.7

Permalink:

http://dlmf.nist.gov/8.7.E3

Encodings:

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Annotations for §8.7 and Ch.8

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8.7.4 $$\gamma (a,x)=\mathrm{\Gamma }\left(a\right)x^{\frac{1}{2}a}{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}{e}_n(-1)x^{\frac{1}{2}n}{I}_{n+a}\left(2x^{1/2}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.7.E4

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Annotations for §8.7 and Ch.8

… ►For an expansion for $\gamma (a,\mathrm{i}x)$ in series of Bessel functions $J_n\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\ge 0$) is small or moderate in magnitude see Barakat (1961).

19: 30.8 Expansions in Series of Ferrers Functions

… ►Then the set of coefficients $a_{n,k}^m({\gamma }^2)$, $k=-R,-R+1,-R+2,\mathrm{\dots }$ is the solution of the difference equation … ►The coefficients $a_{n,k}^m({\gamma }^2)$ satisfy (30.8.4) for all $k$ when we set $a_{n,k}^m({\gamma }^2)=0$ for . …For $k=-N,-N+1,\mathrm{\dots },-R-1$ they are determined from (30.8.4) by forward recursion using $a_{n,-N-1}^m({\gamma }^2)=0$. The set of coefficients $a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2)$, $k=-N-1,-N-2,\mathrm{\dots }$, is the recessive solution of (30.8.4) as $k\to -\mathrm{\infty }$ that is normalized by …It should be noted that if the forward recursion (30.8.4) beginning with $f_{-N-1}=0$, $f_{-N}=1$ leads to $f_{-R}=0$, then $a_{n,k}^m({\gamma }^2)$ is undefined for and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ does not exist. …

20: 5.3 Graphics

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Figure 5.3.1: $\mathrm{\Gamma }\left(x\right)$ and $1/\mathrm{\Gamma }\left(x\right)$. $x_0=1.46\mathrm{\dots }$, $\mathrm{\Gamma }\left(x_0\right)=0.88\mathrm{\dots }$; see §5.4(iii). Magnify

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Figure 5.3.2: $\ln \mathrm{\Gamma }\left(x\right)$. … Magnify

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Figure 5.3.4: $|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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Figure 5.3.5: $1/|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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