normalizing constant

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11: 31.10 Integral Equations and Representations

… ►Here ${\tilde{\kappa }}_m$ is a normalization constant and $C$ is the contour of Example 1. …

12: 18.39 Applications in the Physical Sciences

… ►All are written in the same form as the product of three factors: the square root of a weight function $w(x)$, the corresponding OP or EOP, and constant factors ensuring unit normalization. … ►There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). …

13: 8.3 Graphics

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

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

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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). … ►

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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

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14: 8.2 Definitions and Basic Properties

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8.2.3 $$\gamma (a,z)+\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.3

Referenced by:

§8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

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8.2.6 $${\gamma }^{\ast }(a,z)=z^{-a}P(a,z)=\frac{z^{-a}}{\mathrm{\Gamma }\left(a\right)}\gamma (a,z).$$

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Defines:

${\gamma }^{\ast }(a,z)$: incomplete gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma (a,z)$: incomplete gamma function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.4

Referenced by:

§8.2(iii), §8.6(i)

Permalink:

http://dlmf.nist.gov/8.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

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15: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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16: 30.16 Methods of Computation

… ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd. … ►The coefficients $a_{n,r}^m({\gamma }^2)$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …

17: 8.8 Recurrence Relations and Derivatives

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8.8.5 $$P(a+1,z)=P(a,z)-\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)},$$

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Symbols:

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

A&S Ref:

6.5.21

Referenced by:

§8.25(v)

Permalink:

http://dlmf.nist.gov/8.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.6 $$Q(a+1,z)=Q(a,z)+\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

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Symbols:

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

Referenced by:

§8.25(v), §8.4

Permalink:

http://dlmf.nist.gov/8.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.11 $$P(a+n,z)=P(a,z)-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/8.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.12 $$Q(a+n,z)=Q(a,z)+z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/8.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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18: 31.2 Differential Equations

… ►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ►

§31.2(ii) Normal Form of Heun’s Equation

… ► $w(z)=z^{1-\gamma }{w}_1(z)$ satisfies (31.2.1) if $w_1$ is a solution of (31.2.1) with transformed parameters $q_1=q+(a\delta +ϵ)(1-\gamma )$; ${\alpha }_1=\alpha +1-\gamma$, ${\beta }_1=\beta +1-\gamma$, ${\gamma }_1=2-\gamma$. Next, $w(z)={(z-1)}^{1-\delta }{w}_2(z)$ satisfies (31.2.1) if $w_2$ is a solution of (31.2.1) with transformed parameters $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. …

19: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta ),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

pMML, png, TeX

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

$$\mathrm{\Gamma }\left(a+1\right)\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\pi \mathrm{i}}\mathrm{\Gamma }(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mp \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)+\mathrm{i}T(a,\eta ),$$

was rewritten.

See also:

Annotations for §8.12 and Ch.8

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8.12.15 $$Q(a,a)\sim \frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0)a^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.16 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,a{\mathrm{e}}^{\pm \pi \mathrm{i}})\sim \pm \frac{1}{2}-\frac{\mathrm{i}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0){(-a)}^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $\sin z$: sine function, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Referenced by:

(8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\chi$ , $d(\pm \chi )$ , $A_k(\chi )$ and $B_k(\chi )$

Referenced by:

§8.12, Erratum (V1.0.16) for Equation (8.12.18)

Permalink:

http://dlmf.nist.gov/8.12.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris

See also:

Annotations for §8.12 and Ch.8

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20: 30.8 Expansions in Series of Ferrers Functions

… ►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 …