英国旅行签证存款证明〖办证V信ATV1819〗exp

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11: 5.6 Inequalities

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5.6.5 $$\exp \left((1-s)\psi \left(x+s^{1/2}\right)\right)\le \frac{\mathrm{\Gamma }\left(x+1\right)}{\mathrm{\Gamma }\left(x+s\right)}\le \exp \left((1-s)\psi \left(x+\frac{1}{2}(s+1)\right)\right),$$ .

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Symbols:

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

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E5

Encodings:

pMML, png, TeX

See also:

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

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5.6.9 $$|\mathrm{\Gamma }\left(z\right)|\le {(2\pi )}^{1/2}{|z|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2}\exp \left(\frac{1}{6}{|z|}^{-1}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $y$: real variable and $z$: complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E9

Encodings:

pMML, png, TeX

See also:

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

12: 4.8 Identities

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4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.9 $$\ln \left(\exp z\right)=z,$$ $-\pi \le \mathrm{\Im }z\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.3

Permalink:

http://dlmf.nist.gov/4.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.10 $$\exp \left(\ln z\right)=\exp \left(\mathrm{Ln}z\right)=z.$$

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Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Referenced by:

(25.10.2), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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13: 36.4 Bifurcation Sets

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36.4.11 $$x+\mathrm{i}y=-z^2\exp \left(\frac{2}{3}\mathrm{i}\pi m\right),$$ $m=0,1,2$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $n$: integer and $x$: real parameter

Referenced by:

§36.7(iii)

Permalink:

http://dlmf.nist.gov/36.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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$x=-\frac{1}{12}{z}^2(\exp \left(2\tau \right)\pm 2\exp \left(-\tau \right)),$

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$y=-\frac{1}{12}{z}^2(\exp \left(-2\tau \right)\pm 2\exp \left(\tau \right))$ ,

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14: 10.14 Inequalities; Monotonicity

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10.14.5 $$|J_{\nu }\left(\nu x\right)|\le \frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$ ;

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: exponential function, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.6 $$|J_{\nu }^{\prime }\left(\nu x\right)|\le \frac{{(1+x^2)}^{\frac{1}{4}}}{x{(2\pi \nu )}^{\frac{1}{2}}}\frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$ ;

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.8 $$|J_n\left(nz\right)|\le \frac{\left|z^n\exp \left(n{(1-z^2)}^{\frac{1}{2}}\right)\right|}{{\left|1+{(1-z^2)}^{\frac{1}{2}}\right|}^n},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: exponential function, $n$: integer and $z$: complex variable

A&S Ref:

9.1.63

Permalink:

http://dlmf.nist.gov/10.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14, §10.14 and Ch.10

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15: 27.12 Asymptotic Formulas: Primes

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27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

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

See also:

Annotations for §27.12, §27.12 and Ch.27

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27.12.6 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-d{(\ln x)}^{3/5}{(\ln \ln x)}^{-1/5}\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $d$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E6

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 §27.12, §27.12 and Ch.27

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27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $m$: positive integer and $x$: real number

Referenced by:

Erratum (V1.0.17) for Equation (27.12.8)

Permalink:

http://dlmf.nist.gov/27.12.E8

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes

Change of Notation (effective with 1.0.10):

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

See also:

Annotations for §27.12, §27.12 and Ch.27

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16: 36.15 Methods of Computation

… ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. …

17: 10.56 Generating Functions

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10.56.5 $$\frac{\exp \left(-\sqrt{z^2+2\mathrm{i}zt}\right)}{z}=\frac{{\mathrm{e}}^{-z}}{z}+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{{(-\mathrm{i}t)}^n}{n!}{𝗄}_{n-1}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

18: 19.31 Probability Distributions

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19.31.2 $${\int }_{{ℝ}^n}{({𝐱}^{\mathrm{T}}𝐀𝐱)}^{\mu }\exp \left(-{𝐱}^{\mathrm{T}}𝐁𝐱\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det𝐁}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$ $\mu >-\frac{1}{2}n$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues

Referenced by:

§19.31

Permalink:

http://dlmf.nist.gov/19.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

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19: 32.14 Combinatorics

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32.14.2 $$F(s)=\exp \left(-{\int }_s^{\mathrm{\infty }}(x-s)w^2(x)dx\right),$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $x$: real and $F(s)$: distribution function

Referenced by:

§32.14

Permalink:

http://dlmf.nist.gov/32.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

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20: 36.9 Integral Identities

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36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integral identity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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