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11: 17.3 $q$-Elementary and $q$-Special Functions

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

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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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17.3.6 $${\mathrm{Cos}}_q\left(x\right)=\frac{1}{2}(E_q\left(\mathrm{i}x\right)+E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{q}^{n(2n-1)}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}}.$$

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Defines:

${\mathrm{Cos}}_q\left(x\right)$: $q$-cosine 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.E6

Encodings:

pMML, png, TeX

See also:

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

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12: 17.18 Methods of Computation

… ►For computation of the $q$-exponential function see Gabutti and Allasia (2008). …

13: 20.8 Watson’s Expansions

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20.8.1 $$\frac{{\theta }_2(0,q){\theta }_3(z,q){\theta }_4(z,q)}{{\theta }_2(z,q)}=2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n^2}{\mathrm{e}}^{\mathrm{i}2nz}}{q^{-n}{\mathrm{e}}^{-\mathrm{i}z}+q^n{\mathrm{e}}^{\mathrm{i}z}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.8 and Ch.20

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14: 9.5 Integral Representations

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

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

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

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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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15: 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

16: 7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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17: 4.7 Derivatives and Differential Equations

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§4.7(ii) Exponentials and Powers

… ►The general solution of the differential equation … ►

4.7.13 $$w=\exp \left(\int f(z)dz\right)+\mathrm{constant}.$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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18: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. …

19: 4.14 Definitions and Periodicity

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4.14.1 $$\sin z=\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}},$$

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Defines:

$\sin z$: sine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.1

Referenced by:

§4.45(ii), (9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.2 $$\cos z=\frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2},$$

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Defines:

$\cos z$: cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.2

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.3 $$\cos z\pm \mathrm{i}\sin z={\mathrm{e}}^{\pm \mathrm{i}z},$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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20: 13.26 Addition and Multiplication Theorems

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13.26.2 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{y}{\sqrt{x}}\right)}^n{M}_{\kappa -\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.3 $${\mathrm{e}}^{-\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa -n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.5 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x+y}{x}\right)}^{\mu +\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n}{{\left(1+2\mu \right)}_{n}n!}{\left(\frac{-y}{\sqrt{x}}\right)}^n{M}_{\kappa +\frac{1}{2}n,\mu +\frac{1}{2}n}\left(x\right),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.6 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu +\kappa \right)}_n{y}^n}{n!{(x+y)}^n}{M}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left((y+x)/x\right)>\frac{1}{2}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§13.26(i)

Permalink:

http://dlmf.nist.gov/13.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(i), §13.26 and Ch.13

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13.26.12 $${\mathrm{e}}^{\frac{1}{2}y}{\left(\frac{x}{x+y}\right)}^{\kappa }\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left(\frac{-y}{x+y}\right)}^n{W}_{\kappa +n,\mu }\left(x\right),$$ $\mathrm{\Re }\left(y/x\right)>-\frac{1}{2}$.

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.26.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.26(ii), §13.26 and Ch.13

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