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31: 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

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

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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

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

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8.10.3 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)=1+\frac{a-1}{x}\vartheta ,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/8.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

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8.10.5 $x>0$, ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer, $A_n$: minima and $B_n$: maxima

Permalink:

http://dlmf.nist.gov/8.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10, §8.10 and Ch.8

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8.10.11 $${(1-{\mathrm{e}}^{-{\alpha }_{a}x})}^a\le P(a,x)\le {(1-{\mathrm{e}}^{-{\beta }_{a}x})}^a,$$ $x\ge 0$, $a>0$,

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Symbols:

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

Referenced by:

(7.8.8), §8.10

Permalink:

http://dlmf.nist.gov/8.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10, §8.10 and Ch.8

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32: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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33: 20.7 Identities

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20.7.16 $${\theta }_1\left(2z\right|2\tau )=A{\theta }_1\left(z\right|\tau ){\theta }_2\left(z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex, $\tau$: lattice parameter and $A$

Permalink:

http://dlmf.nist.gov/20.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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20.7.19 $${\theta }_4\left(2z\right|2\tau )=A{\theta }_3\left(z\right|\tau ){\theta }_4\left(z\right|\tau ).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex, $\tau$: lattice parameter and $A$

Permalink:

http://dlmf.nist.gov/20.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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§20.7(viii) Transformations of Lattice Parameter

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20.7.28 $${\theta }_3\left(z\right|\tau +1)={\theta }_4\left(z\right|\tau ),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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20.7.29 $${\theta }_4\left(z\right|\tau +1)={\theta }_3\left(z\right|\tau ).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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34: 11.7 Integrals and Sums

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11.7.13 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_0\left(t\right)dt=\frac{2}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve 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, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_1\left(t\right)dt=\frac{2}{\pi a}-\frac{2a}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve 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, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐋}_0\left(t\right)dt=\frac{2}{\pi \sqrt{a^2-1}}\arcsin \left(\frac{1}{a}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arcsin z$: arcsine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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35: 8.28 Software

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§8.28(ii) Incomplete Gamma Functions for Real Argument and Parameter

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§8.28(iii) Incomplete Gamma Functions for Complex Argument and Parameter

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§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters

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§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters

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36: 8.8 Recurrence Relations and Derivatives

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8.8.3 $$w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$$

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Symbols:

$z$: complex variable, $a$: parameter and $w(a,z)$: function

Permalink:

http://dlmf.nist.gov/8.8.E3

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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8.8.15 $$\frac{d^n}{{dz}^n}(z^{-a}\gamma (a,z))={(-1)}^n{z}^{-a-n}\gamma (a+n,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.16 $$\frac{d^n}{{dz}^n}(z^{-a}\mathrm{\Gamma }(a,z))={(-1)}^n{z}^{-a-n}\mathrm{\Gamma }(a+n,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

A&S Ref:

6.5.26

Referenced by:

§8.19(v)

Permalink:

http://dlmf.nist.gov/8.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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37: 30.4 Functions of the First Kind

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30.4.1 $${\int }_{-1}^1{\left({\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right)}^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.11

Referenced by:

§30.4(iii)

Permalink:

http://dlmf.nist.gov/30.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

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30.4.3 $${\mathrm{𝖯𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2).$$

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Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(ii), §30.4 and Ch.30

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30.4.7 $$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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30.4.8 $$c_n=(n+\frac{1}{2})\frac{(n-m)!}{(n+m)!}{\int }_{-1}^{1}f(t){\mathrm{𝖯𝗌}}_n^m(t,{\gamma }^2)dt.$$

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Defines:

$c_n$: coefficient (locally)

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and $f(x)$: function

Permalink:

http://dlmf.nist.gov/30.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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30.4.9 $$\underset{N\to \mathrm{\infty }}{lim}{\int }_{-1}^1{\left|f(x)-\sum _{n=m}^N{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right|}^{2}dx=0.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Permalink:

http://dlmf.nist.gov/30.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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38: 13.2 Definitions and Basic Properties

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Kummer’s Equation

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13.2.11 $$U(a,-n,z)=z^{n+1}U(a+n+1,n+2,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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39: 9.13 Generalized Airy Functions

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9.13.15

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Symbols:

$A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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9.13.16

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Symbols:

$B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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9.13.20 $$U_1(x,\alpha )=\frac{1}{{(\alpha +2)}^{1/(\alpha +2)}}\mathrm{\Gamma }\left(\frac{\alpha +1}{\alpha +2}\right)x^{1/2}{J}_{-1/(\alpha +2)}\left(\frac{2}{\alpha +2}{x}^{(\alpha +2)/2}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\alpha$: parameter, $x$: parameter and $U_{\mathrm{i}}$: Smirnov’s generalized Airy function

Source:

Smirnov (1960, (2.10), p. 10)

Referenced by:

1st item

Permalink:

http://dlmf.nist.gov/9.13.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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9.13.21 $$U_2(x,\alpha )={(\alpha +2)}^{1/(\alpha +2)}\mathrm{\Gamma }\left(\frac{\alpha +3}{\alpha +2}\right)x^{1/2}{J}_{1/(\alpha +2)}\left(\frac{2}{\alpha +2}{x}^{(\alpha +2)/2}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\alpha$: parameter, $x$: parameter and $U_{\mathrm{i}}$: Smirnov’s generalized Airy function

Source:

Smirnov (1960, (2.11), p. 11)

Referenced by:

1st item

Permalink:

http://dlmf.nist.gov/9.13.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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9.13.32 $$f(p-3)-zf(p-1)+(p-1)f(p)=0.$$

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Defines:

$f$: function (locally)

Symbols:

$z$: complex variable and $p$: parameter

Source:

Reid (1972, (A9), p. 363)

Permalink:

http://dlmf.nist.gov/9.13.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(ii), §9.13 and Ch.9

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40: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

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Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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Barrett’s Expansions

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Dunster’s Approximations

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