英国旅行签证存款证明〖办证V信ATV1819〗exp

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1: 17.3 $q$-Elementary and $q$-Special Functions

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17.3.1 $$e_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{x}^n}{{(q;q)}_n}=\frac{1}{{((1-q)x;q)}_{\mathrm{\infty }}},$$

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Defines:

$e_q\left(x\right)$: $q$-exponential function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.2 $$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$

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Defines:

$E_q\left(x\right)$: $q$-exponential function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.3 $${\sin }_q\left(x\right)=\frac{1}{2\mathrm{i}}(e_q\left(\mathrm{i}x\right)-e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}},$$

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Defines:

${\sin }_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.4 $${\mathrm{Sin}}_q\left(x\right)=\frac{1}{2\mathrm{i}}(E_q\left(\mathrm{i}x\right)-E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{q}^{n(2n+1)}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}}.$$

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Defines:

${\mathrm{Sin}}_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.5 $${\cos }_q\left(x\right)=\frac{1}{2}(e_q\left(\mathrm{i}x\right)+e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}},$$

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Defines:

${\cos }_q\left(x\right)$: $q$-cosine function

Symbols:

$\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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2: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

3: 4.2 Definitions

… ►The function $\exp$ is an entire function of $z$, with no real or complex zeros. … ►

4.2.20 $$\exp \left(z+2\pi \mathrm{i}\right)=\exp z.$$

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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 and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.21 $$\exp \left(-z\right)=1/\exp \left(z\right).$$

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Symbols:

$\exp z$: exponential function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.22 $$|\exp z|=\exp \left(\mathrm{\Re }z\right).$$

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Symbols:

$\exp z$: exponential function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.2.13

Permalink:

http://dlmf.nist.gov/4.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.28 $$z^a=\exp \left(a\ln z\right).$$

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Symbols:

$\exp z$: exponential function, $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.7

Permalink:

http://dlmf.nist.gov/4.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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4: 36.11 Leading-Order Asymptotics

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36.11.3

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.4

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $o\left(x\right)$: order less than and $x$: real parameter

Referenced by:

§36.11

Permalink:

http://dlmf.nist.gov/36.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

5: 36.2 Catastrophes and Canonical Integrals

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36.2.6 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right){\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,$$

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Symbols:

$\pi$: 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, $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.2(i), §36.2(ii), §36.5(iii)

Permalink:

http://dlmf.nist.gov/36.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)={\displaystyle \frac{4\pi }{3^{1/3}}}\exp \left(\mathrm{i}\left(\frac{2}{27}{z}^3-\frac{1}{3}xz\right)\right)\left(\exp \left(-\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{+}(𝐱)+\exp \left(\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{-}(𝐱)\right),$

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${\mathrm{F}}_{\pm }(𝐱)={\displaystyle {\int }_0^{\mathrm{\infty }}}\cos \left(ry\exp \left(\pm \mathrm{i}{\displaystyle \frac{\pi }{6}}\right)\right)\exp \left(2\mathrm{i}{r}^{2}z\exp \left(\pm \mathrm{i}{\displaystyle \frac{\pi }{3}}\right)\right)\mathrm{Ai}\left(3^{2/3}{r}^2+3^{-1/3}\exp \left(\mp \mathrm{i}{\displaystyle \frac{\pi }{3}}\right)\left(\frac{1}{3}{z}^2-x\right)\right)dr.$

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36.2.9 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)=\frac{2\pi }{3^{1/3}}{\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/6\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/6\right)}\exp \left(\mathrm{i}(s^3+xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)ds.$$

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Symbols:

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

Referenced by:

§36.2(i), §36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

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36.2.12 $${\mathrm{\Psi }}_0=\sqrt{\pi }\exp \left(\mathrm{i}\frac{\pi }{4}\right).$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function and $\mathrm{i}$: imaginary unit

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

…

6: 9.5 Integral Representations

… ►

9.5.3 $$\mathrm{Bi}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3+xt\right)dt+\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\sin \left(\frac{1}{3}{t}^3+xt\right)dt.$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b]$: half-closed interval, $\sin z$: sine function and $x$: real variable

Proof sketch:

Use (9.5.5) with the substitions: for the paths $(-\mathrm{\infty },0]$ use $t=-\tau$, for the paths $[0,\mathrm{\infty }{\mathrm{e}}^{\pm \pi \mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm \pi \mathrm{i}/3}\tau$. For the resulting integrals use (4.14.1) and (4.14.2).

A&S Ref:

10.4.33

Referenced by:

(9.12.19)

Permalink:

http://dlmf.nist.gov/9.5.E3

Encodings:

pMML, png, TeX

See also:

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

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9.5.4 $$\mathrm{Ai}\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt,$$

ⓘ

Symbols:

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

Source:

Olver (1997b, p. 53)

Referenced by:

§36.2(ii), §9.14, §9.17(iii), (9.5.5)

Permalink:

http://dlmf.nist.gov/9.5.E4

Encodings:

pMML, png, TeX

See also:

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

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9.5.5 $$\mathrm{Bi}\left(z\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt+\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }{\mathrm{e}}^{-\pi \mathrm{i}/3}}\exp \left(\frac{1}{3}{t}^3-zt\right)dt.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.2.10) and (9.5.4).

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/9.5.E5

Encodings:

pMML, png, TeX

See also:

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

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9.5.6 $$\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right)dt,$$ .

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Symbols:

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

Source:

Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$)

Referenced by:

Erratum (V1.0.15) for Equation (9.5.6)

Permalink:

http://dlmf.nist.gov/9.5.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

The validity constraint was added.

See also:

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

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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

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7: 18.32 OP’s with Respect to Freud Weights

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18.32.1 $$w(x)=\exp \left(-Q(x)\right),$$ ,

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Symbols:

$\exp z$: exponential function, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32

Permalink:

http://dlmf.nist.gov/18.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.32 and Ch.18

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18.32.2 $$w(x)={\left|x\right|}^{\alpha }\exp \left(-Q(x)\right),$$ $x\in ℝ$, $\alpha >-1$,

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Symbols:

$\in$: element of, $\exp z$: exponential function, $ℝ$: real line, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32, Erratum (V1.2.0) Section 18.32

Permalink:

http://dlmf.nist.gov/18.32.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.32 and Ch.18

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8: 20.6 Power Series

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20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\delta }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.4 $${\theta }_3\left(\pi z\right|\tau )={\theta }_3\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\beta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\beta }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.5 $${\theta }_4\left(\pi z\right|\tau )={\theta }_4\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\gamma }_{2j}(\tau )z^{2j}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\gamma }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

…

9: 32.15 Orthogonal Polynomials

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32.15.1 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{4}{\xi }^4-z{\xi }^2\right)p_m(\xi )p_n(\xi )d\xi ={\delta }_{m,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $m$: integer, $n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

…

10: 4.12 Generalized Logarithms and Exponentials

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4.12.7 $$\varphi (x)=\underset{\lfloor x\rfloor \text{ times}}{\underbrace{\exp \mathrm{\cdots }\exp }}(x-\lfloor x\rfloor ),$$ $x>1$.

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Symbols:

$\exp z$: exponential function, $\lfloor x\rfloor$: floor of $x$ , $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

…