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21: 23.6 Relations to Other Functions

… ►With $z=\pi u/(2{\omega }_1)$, …

22: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

…

23: 9.5 Integral Representations

… ►

9.5.7 $$\mathrm{Ai}\left(z\right)=\frac{{\mathrm{e}}^{-\zeta }}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-z^{1/2}{t}^2\right)\cos \left(\frac{1}{3}{t}^3\right)dt,$$ .

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $\zeta$: change of variable

Source:

Copson (1963, (2.1))

Referenced by:

§9.5(ii)

Permalink:

http://dlmf.nist.gov/9.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.8 $$\mathrm{Ai}\left(z\right)=\frac{{\mathrm{e}}^{-\zeta }{\zeta }^{-1/6}}{\sqrt{\pi }{(48)}^{1/6}\mathrm{\Gamma }\left(\frac{5}{6}\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-1/6}{\left(2+\frac{t}{\zeta }\right)}^{-1/6}dt,$$ .

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{ph}$: phase, $z$: complex variable and $\zeta$: change of variable

Source:

Follows from (9.6.21) and (13.4.4).

Referenced by:

§9.17(iii), §9.5(ii)

Permalink:

http://dlmf.nist.gov/9.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

…

24: 27.2 Functions

… ►Gauss and Legendre conjectured that $\pi \left(x\right)$ is asymptotic to $x/\ln x$ as $x\to \mathrm{\infty }$: ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding $x$ and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.4 $$p_n\sim n\ln n.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

ⓘ

Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

…

25: 19.25 Relations to Other Functions

… ►then the five nontrivial permutations of $x,y,z$ that leave $R_F$ invariant change $k^2$ ($=(z-y)/(z-x)$) into $1/k^2$, $k^{\prime }{}^2$, $1/k^{\prime }{}^2$, $-k^2/k^{\prime }{}^2$, $-k^{\prime }{}^2/k^2$, and $\sin \varphi$ ($=\sqrt{(z-x)/z}$) into $k\sin \varphi$, $-\mathrm{i}\tan \varphi$, $-\mathrm{i}{k}^{\prime }\tan \varphi$, $(k^{\prime }\sin \varphi )/\sqrt{1-k^2{\sin }^2\varphi }$, $-\mathrm{i}k\sin \varphi /\sqrt{1-k^2{\sin }^2\varphi }$. …

26: Errata

… ►

To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

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

…

27: 2.5 Mellin Transform Methods

… ►

2.5.9 $$ℳf\left(1-z\right)=\frac{\pi }{\sin \left(\pi z\right)},$$ ,

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\sin z$: sine function and $f(t)=1/(1+t)$: function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

►

2.5.10 $$ℳh\left(z\right)=\frac{2^{z-1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}z\right)}{{\mathrm{\Gamma }}^2\left(1-\frac{1}{2}z\right)\mathrm{\Gamma }\left(1+\nu -\frac{1}{2}z\right)\mathrm{\Gamma }\left(z\right)}\frac{\pi }{\sin \left(\pi z\right)},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\sin z$: sine function, $h(t)=J_{\nu }^2\left(t\right)$: function and $f(t)=1/(1+t)$: function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

… ►

2.5.11 $${res}_{z=n}\left[x^{-z}ℳf\left(1-z\right)ℳh\left(z\right)\right]=(a_n\ln x+b_n)x^{-n},$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\ln z$: principal branch of logarithm function, $res$: residue, $h(t)=J_{\nu }^2\left(t\right)$: function, $a_n$: coefficients, $b_n$: coefficients and $f(t)=1/(1+t)$: function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E11

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

…

28: 1.14 Integral Transforms

… ►

1.14.17 $$ℒ\left(f\right)\left(s\right)=ℒf\left(s\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}f(t)dt.$$

ⓘ

Defines:

$ℒ\left(f\right)\left(s\right)$: Laplace transform

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Keywords:

Laplace transform

A&S Ref:

29.1.1

Permalink:

http://dlmf.nist.gov/1.14.E17

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒ\left(f\right)\left(s\right)$ or $ℒf\left(s\right)$ from $ℒ(f(t);s)$.

See also:

Annotations for §1.14(iii), §1.14 and Ch.1

… ►

1.14.19 $$ℒf\left(s\right)\to 0,$$ $\mathrm{\Re }s\to \mathrm{\infty }$.

ⓘ

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

Permalink:

http://dlmf.nist.gov/1.14.E19

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(s\right)$ from $ℒ(f(t);s)$.

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►If also ${lim}_{t\to 0+}f(t)/t$ exists, then … ►where $A_p=\tan \left(\frac{1}{2}\pi /p\right)$ when , or $\cot \left(\frac{1}{2}\pi /p\right)$ when $p\ge 2$. … ►

1.14.46 $$\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathscr{H}f\left(u\right){\mathrm{e}}^{\mathrm{i}ux}du=-\mathrm{i}(\mathrm{sign}x)ℱf\left(x\right),$$

ⓘ

Symbols:

$ℱ\left(f\right)\left(s\right)$: Fourier transform, $\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{sign}x$: sign of $x$

Referenced by:

§1.14(v)

Permalink:

http://dlmf.nist.gov/1.14.E46

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Hilbert transform was changed to $\mathscr{H}f\left(t\right)$ from $\mathscr{H}(f;t)$, and the notation for the Fourier transform was changed to $ℱf\left(x\right)$ from $F(x)$. In addition, the integration variable was changed from $t$ to $u$.

See also:

Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

…

29: 15.12 Asymptotic Approximations

… ►

15.12.5 $$\mathbf{F}(\genfrac{}{}{0pt}{}{a+\lambda ,b-\lambda }{c};\frac{1}{2}-\frac{1}{2}z)=2^{(a+b-1)/2}\frac{{(z+1)}^{(c-a-b-1)/2}}{{(z-1)}^{c/2}}\sqrt{\zeta \sinh \zeta }{\left(\lambda +\frac{1}{2}a-\frac{1}{2}b\right)}^{1-c}\left(I_{c-1}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)(1+O({\lambda }^{-2}))+\frac{I_{c-2}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)}{2\lambda +a-b}\left(\left(c-\frac{1}{2}\right)\left(c-\frac{3}{2}\right)\left(\frac{1}{\zeta }-\coth \zeta \right)+\frac{1}{2}(2c-a-b-1)(a+b-1)\tanh \left(\frac{1}{2}\zeta \right)+O({\lambda }^{-2})\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}\mathbf{F}(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.11 $$\beta ={\left(-\frac{3}{2}\zeta +\frac{9}{4}\ln \left(\frac{2+{\mathrm{e}}^{\zeta }}{2+{\mathrm{e}}^{-\zeta }}\right)\right)}^{1/3},$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $\zeta$: change of variable and $\beta$

Permalink:

http://dlmf.nist.gov/15.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.13 $$G_0(\pm \beta )={\left(2+{\mathrm{e}}^{\pm \zeta }\right)}^{c-b-(1/2)}{\left(1+{\mathrm{e}}^{\pm \zeta }\right)}^{a-c+(1/2)}{\left(z-1-{\mathrm{e}}^{\pm \zeta }\right)}^{-a+(1/2)}\sqrt{\frac{\beta }{{\mathrm{e}}^{\zeta }-{\mathrm{e}}^{-\zeta }}}.$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $G_0(\pm \zeta )$: function

Permalink:

http://dlmf.nist.gov/15.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

…

30: 19.2 Definitions

… ►The cases with $\varphi =\pi /2$ are the complete integrals: … ►

$k_c=k^{\prime },$

►

$p=1-{\alpha }^2,$

►

$x=\tan \varphi ,$

… ►If , then $k_c$ is pure imaginary. …