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21: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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22: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
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23: 13.8 Asymptotic Approximations for Large Parameters

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13.8.9 $$M(a,b,x)=\mathrm{\Gamma }\left(b\right){\mathrm{e}}^{\frac{1}{2}x}{\left((\frac{1}{2}b-a)x\right)}^{\frac{1}{2}-\frac{1}{2}b}\left(J_{b-1}\left(\sqrt{2x(b-2a)}\right)+\mathrm{env}{J}_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a\right|}^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

13.5.13 (in different form)

Referenced by:

§13.9(i)

Permalink:

http://dlmf.nist.gov/13.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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13.8.10 $$U(a,b,x)=\mathrm{\Gamma }\left(\frac{1}{2}b-a+\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}x}{x}^{\frac{1}{2}-\frac{1}{2}b}\left(\cos \left(a\pi \right)J_{b-1}\left(\sqrt{2x(b-2a)}\right)-\sin \left(a\pi \right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right)+\mathrm{env}{Y}_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a\right|}^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function and $x$: real variable

A&S Ref:

13.5.15 (in different form)

Referenced by:

§13.9(ii)

Permalink:

http://dlmf.nist.gov/13.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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24: 19.12 Asymptotic Approximations

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19.12.1 $$K\left(k\right)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}k_{}^{\prime }{}_{}{}^{2m}\left(\ln \left(\frac{1}{k^{\prime }}\right)+d(m)\right),$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $d(m)$: function

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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19.12.2 $$E\left(k\right)=1+\frac{1}{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{3}{2}\right)}_m}{{\left(2\right)}_{m}m!}k_{}^{\prime }{}_{}{}^{2m+2}\left(\ln \left(\frac{1}{k^{\prime }}\right)+d(m)-\frac{1}{(2m+1)(2m+2)}\right),$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $d(m)$: function

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

… ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. … ►

19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

25: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►with diffusion constant $\alpha =\pi /4$. …These two apparently different solutions differ only in their normalization and boundary conditions. … ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

26: 20.1 Special Notation

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$m$, $n$	integers.
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$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)

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27: 28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28: 9.8 Modulus and Phase

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9.8.8 $$\varphi \left(x\right)=\arctan \left({\mathrm{Ai}}^{\prime }\left(x\right)/{\mathrm{Bi}}^{\prime }\left(x\right)\right).$$

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Defines:

$\varphi \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.09–2.10), p. 396); with different notation

Referenced by:

(9.11.19), (9.8.12), (9.8.16), (9.8.17)

Permalink:

http://dlmf.nist.gov/9.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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29: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

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30: 9.2 Differential Equation

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9.2.5 $$\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}=\mathrm{0.61492\hspace{0.33em}66274}\mathrm{\dots },$$

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Symbols:

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

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.4 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

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9.2.6 $${\mathrm{Bi}}^{\prime }\left(0\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}=\mathrm{0.44828\hspace{0.33em}83573}\mathrm{\dots }.$$

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Symbols:

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

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.5 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

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