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11: 4.34 Derivatives and Differential Equations

… ►

4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

►

4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

►

4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

ⓘ

Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

…

12: 31.6 Path-Multiplicative Solutions

… ►with $(s_1,s_2)\in \{0,1,a\}$, but with another set of $\{q_m\}$. This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the $z$-plane that encircles $s_1$ and $s_2$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. …

13: 4.16 Elementary Properties

… ►

Table 4.16.3: Trigonometric functions: interrelations. …

► ►►►►►►►

	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
…
$\cos \theta$	${(1-a^2)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a^{-1}$	$a{(1+a^2)}^{-1/2}$
$\tan \theta$	$a{(1-a^2)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	${(a^2-1)}^{-1/2}$	${(a^2-1)}^{1/2}$	$a^{-1}$
$\csc \theta$	$a^{-1}$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a$	$a{(a^2-1)}^{-1/2}$	${(1+a^2)}^{1/2}$
$\sec \theta$	${(1-a^2)}^{-1/2}$	$a^{-1}$	${(1+a^2)}^{1/2}$	$a{(a^2-1)}^{-1/2}$	$a$	$a^{-1}{(1+a^2)}^{1/2}$
$\cot \theta$	$a^{-1}{(1-a^2)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	${(a^2-1)}^{1/2}$	${(a^2-1)}^{-1/2}$	$a$

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14: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …

15: 17.11 Transformations of $q$-Appell Functions

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17.11.1 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\frac{{(a,bx,b^{\prime }y;q)}_{\mathrm{\infty }}}{{(c,x,y;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,x,y}{bx,b^{\prime }y};q,a),$$

ⓘ

Symbols:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2)

Permalink:

http://dlmf.nist.gov/17.11.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

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17.11.3 $${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)=\frac{{(a,bx;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a^{\prime },b^{\prime };q)}_n{(x;q)}_r{(c/a;q)}_{n+r}{a}^r{y}^n}{{(q,c/a;q)}_n{(q,bx;q)}_r}.$$

ⓘ

Symbols:

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§17.11, Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.11.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.11 and Ch.17

… ►

17.11.4 $$\sum _{m_1,\mathrm{\dots },m_n\geqq 0}\frac{{(a;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}{(b_1;q)}_{m_1}{(b_2;q)}_{m_2}\mathrm{\cdots }{(b_n;q)}_{m_n}{x}_1^{m_1}{x}_2^{m_2}\mathrm{\cdots }{x}_n^{m_n}}{{(q;q)}_{m_1}{(q;q)}_{m_2}\mathrm{\cdots }{(q;q)}_{m_n}{(c;q)}_{m_1+m_2+\mathrm{\cdots }+m_n}}=\frac{{(a,b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n;q)}_{\mathrm{\infty }}}{{(c,x_1,x_2,\mathrm{\dots },x_n;q)}_{\mathrm{\infty }}}{}_{n+1}{}^{}\varphi _n^{}(\genfrac{}{}{0pt}{}{c/a,x_1,x_2,\mathrm{\dots },x_n}{b_1{x}_1,b_2{x}_2,\mathrm{\dots },b_n{x}_n};q,a).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.11 and Ch.17

16: 4.20 Derivatives and Differential Equations

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4.20.13 $$w=(1/a)\sin \left(az+c\right),$$

ⓘ

Symbols:

$\sin z$: sine function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.14 $$w=(1/a)\tan \left(az+c\right),$$

ⓘ

Symbols:

$\tan z$: tangent function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

…

17: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

8.18.8 $$x_0=a/(a+b).$$

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Symbols:

$a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(ii)

Permalink:

http://dlmf.nist.gov/8.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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8.18.9 $$I_x(a,b)\sim \frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi (a+b)}}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{c}_k(\eta )}{{(a+b)}^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

►uniformly for $x\in (0,1)$ and $a/(a+b)$, $b/(a+b)\in [\delta ,1-\delta ]$, where $\delta$ again denotes an arbitrary small positive constant. … ►Let $\mu =b/a$, and $x_0$ again be as in (8.18.8). … ►

8.18.14 $$I_x(a,b)\sim Q(b,a\zeta )-\frac{{(2\pi b)}^{-1/2}}{{\mathrm{\Gamma }}^{\ast }\left(b\right)}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{h_k(\zeta ,\mu )}{a^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $Q(a,z)$: normalized incomplete gamma function, ${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $\mu$ , $\zeta$: variable and $h_k(\zeta ,\mu )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

…

18: 17.2 Calculus

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17.2.2 $${(a;q)}_{-n}=\frac{1}{{(a{q}^{-n};q)}_n}=\frac{{(-q/a)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}}{{(q/a;q)}_n}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.10 $$\frac{{(a;q)}_n}{{(b;q)}_n}=\frac{{(q^{1-n}/a;q)}_n}{{(q^{1-n}/b;q)}_n}{\left(\frac{a}{b}\right)}^n,$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.12 $$\frac{{(a{q}^{-n};q)}_n}{{(b{q}^{-n};q)}_n}=\frac{{(q/a;q)}_n}{{(q/b;q)}_n}{\left(\frac{a}{b}\right)}^n.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.15 $${(a{q}^{-n};q)}_k=\frac{{(a;q)}_k{(q/a;q)}_n}{{(q^{1-k}/a;q)}_n}{q}^{-nk},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

►

17.2.16 $${(a{q}^{-n};q)}_{n-k}=\frac{{(q/a;q)}_n}{{(q/a;q)}_k}{\left(-\frac{a}{q}\right)}^{n-k}{q}^{\left(\genfrac{}{}{0.0pt}{}{k}{2}\right)-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(i), §17.2 and Ch.17

…

19: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

$\lambda =z/a,$

… ►

8.12.3 $$P(a,z)=\frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{a/2}\right)-S(a,\eta ),$$

ⓘ

Symbols:

$\mathrm{erfc}z$: complementary error function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

►

8.12.4 $$Q(a,z)=\frac{1}{2}\mathrm{erfc}\left(\eta \sqrt{a/2}\right)+S(a,\eta ),$$

ⓘ

Symbols:

$\mathrm{erfc}z$: complementary error function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.6 $$z^{-a}{\gamma }^{\ast }(-a,-z)=\cos \left(\pi a\right)-2\sin \left(\pi a\right)\left(\frac{{\mathrm{e}}^{\frac{1}{2}a{\eta }^2}}{\sqrt{\pi }}F\left(\eta \sqrt{a/2}\right)+T(a,\eta )\right),$$

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

$\chi =(z-a)/\sqrt{z},$

…

20: 31.3 Basic Solutions

… ►Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. … ►

31.3.12 $$H\mathrm{\ell }(1/a,q/a;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma -\delta ;z/a),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(iii), §31.3 and Ch.31

►which arises from the homography $\tilde{z}=z/a$, and to …