Catalan constant

SearchAdvancedHelp

(0.002 seconds)

21—30 of 438 matching pages

21: 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).

22: 8.7 Series Expansions

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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).

23: 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. …

24: 5.3 Graphics

… ►

►►

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

►

►►

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

… ►

►

►

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

►

►

►

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

…

25: 16.13 Appell Functions

… ►

16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

ⓘ

Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►

16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

ⓘ

Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

►Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and ${\gamma }^{\prime }$ is a nonpositive integer. …

26: 5.1 Special Notation

… ► ►►►►

$j,m,n$	nonnegative integers.
…
$\delta$	arbitrary small positive constant.
$\gamma$	Euler’s constant (§5.2(ii)).
…

►The main functions treated in this chapter are the gamma function $\mathrm{\Gamma }\left(z\right)$, the psi function (or digamma function) $\psi \left(z\right)$, the beta function $\mathrm{B}(a,b)$, and the $q$-gamma function ${\mathrm{\Gamma }}_q\left(z\right)$. ►The notation $\mathrm{\Gamma }\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\mathrm{\Pi }(z-1)$ (Gauss) and $(z-1)!$. …

27: 5.6 Inequalities

… ►

5.6.2 $$\frac{1}{\mathrm{\Gamma }\left(x\right)}+\frac{1}{\mathrm{\Gamma }\left(1/x\right)}\le 2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6 and Ch.5

►

5.6.3 $$\frac{1}{{(\mathrm{\Gamma }\left(x\right))}^2}+\frac{1}{{(\mathrm{\Gamma }\left(1/x\right))}^2}\le 2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6 and Ch.5

… ►

5.6.4 .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $s$: real or complex variable and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

… ►

5.6.6 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$

ⓘ

Symbols:

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

A&S Ref:

6.1.26

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

… ►

5.6.8 $$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

…

28: 8.3 Graphics

… ►

►►

Figure 8.3.1: $\mathrm{\Gamma }(a,x)$, $a$ = 0. … Magnify

►

►►

Figure 8.3.2: $\gamma (a,x)$, $a$ = 0. … Magnify

►

►►

Figure 8.3.3: $\gamma (a,x)$, $a$ = 1, 2, 2. … Magnify

… ►Some monotonicity properties of ${\gamma }^{\ast }(a,x)$ and $\mathrm{\Gamma }(a,x)$ in the four quadrants of the ($a,x$)-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). … ►

►

►

Figure 8.3.8: $\mathrm{\Gamma }(0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. …When $x=y=0$, $\mathrm{\Gamma }(0.25,0)=\mathrm{\Gamma }\left(0.25\right)=3.625\mathrm{\dots }$. Magnify 3D Help

…

29: 16.1 Special Notation

… ► ►►►

$p,q$	nonnegative integers.
…
$\delta$	arbitrary small positive constant.
…

►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.

30: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

$$\gamma ,\delta >-1,\beta >N+\gamma .$$

►

… ►

►The first four sets imply $\gamma +\delta >-2$, and the last four imply . …