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1: 8.21 Generalized Sine and Cosine Integrals

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8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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2: 4.9 Continued Fractions

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4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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4.9.2 $$\ln \left(\frac{1+z}{1-z}\right)=\frac{2z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\frac{16z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.40

Permalink:

http://dlmf.nist.gov/4.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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$${\mathrm{e}}^z=\frac{1}{1-}\frac{z}{1+}\frac{z}{2-}\frac{z}{3+}\frac{z}{2-}\frac{z}{5+}\frac{z}{2-}\mathrm{\cdots }$$

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$$=1+\frac{z}{1-}\frac{z}{2+}\frac{z}{3-}\frac{z}{2+}\frac{z}{5-}\frac{z}{2+}\frac{z}{7-}\mathrm{\cdots }$$

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$$=1+\frac{z}{1-(z/2)+}\frac{z^2/(4\cdot 3)}{1+}\frac{z^2/(4\cdot 15)}{1+}\frac{z^2/(4\cdot 35)}{1+}\mathrm{\cdots }\frac{z^2/(4(4n^2-1))}{1+}\mathrm{\cdots }$$

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3: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /6$	$\frac{1}{2}$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{3}\sqrt{3}$	$2$	$\frac{2}{3}\sqrt{3}$	$\sqrt{3}$
…

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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

4: 10.10 Continued Fractions

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10.10.1 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}-}\frac{1}{2(\nu +1)z^{-1}-}\frac{1}{2(\nu +2)z^{-1}-}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

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

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

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10.10.2 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1-}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1-}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1-}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

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

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

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5: 10.33 Continued Fractions

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10.33.1 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}+}\frac{1}{2(\nu +1)z^{-1}+}\frac{1}{2(\nu +2)z^{-1}+}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

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10.33.2 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1+}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1+}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1+}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

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6: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.4 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{e,f}{aq/c,aq/d};q,\frac{aq}{ef})=\frac{{(q/c,q/d,aq/e,aq/f;q)}_{\mathrm{\infty }}}{{(aq,q/a,aq/(cd),aq/(ef);q)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(1-a{q}^{2n}){(c,d,e,f;q)}_n}{(1-a){(aq/c,aq/d,aq/e,aq/f;q)}_n}{\left(\frac{q{a}^3}{cdef}\right)}^n{q}^{n^2}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.5 $$\frac{{(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)}_{\mathrm{\infty }}}{{(fa,ga,f/a,g/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^2}{bcdefg})=\frac{{(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)}_{\mathrm{\infty }}}{{(fa,q/(fa),aq/f,f/a,g/f,fg,q{f}^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{f^2,qf,-qf,fb,fc,fd,fe,fg}{f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^2}{bcdefg})+\mathrm{idem}(f;g).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

►

17.10.6 $$\frac{{(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q)}_{\mathrm{\infty }}}{{(ag,ah,ak,g/a,h/a,k/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\psi _{10}^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga,ha,ka}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^3}{bcdefghk})=\frac{{(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q)}_{\mathrm{\infty }}}{{(gh,gk,h/g,k/g,ag,q/(ag),g/a,aq/g,q{g}^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\varphi _9^{}(\genfrac{}{}{0pt}{}{g^2,qg,-qg,gb,gc,gd,ge,gf,gh,gk}{g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,\frac{q^3}{bcdefghk})+\mathrm{idem}(g;h,k).$$

ⓘ

Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ function, ${(a;q)}_n$: $q$-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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer and $q$: complex base

Referenced by:

Erratum (V1.2.4) for Equation (17.10.6)

Permalink:

http://dlmf.nist.gov/17.10.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.2.4):

In the numerator of the argument of the basic bilateral hypergeometric function ${}_{10}{}^{}\psi _{10}^{}$ and in the numerator of the arguments of the basic hypergeometric ${}_{10}{}^{}\varphi _9^{}$ functions, we replaced $q^2$ by $q^3$. We also added a missing factor $1/{(k/g;q)}_{\mathrm{\infty }}$ in the first term on the right-hand side.

See also:

Annotations for §17.10, §17.10 and Ch.17

7: 18.13 Continued Fractions

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18.13.1 $$\frac{-1}{x+}\frac{-1}{2x+}\frac{-1}{2x+}\mathrm{\cdots },$$

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Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.2 $$\frac{-1}{2x+}\frac{-1}{2x+}\frac{-1}{2x+}\mathrm{\cdots }.$$

ⓘ

Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.3 $$\frac{a_1}{x+}\frac{-\frac{1}{2}}{\frac{3}{2}x+}\frac{-\frac{2}{3}}{\frac{5}{3}x+}\frac{-\frac{3}{4}}{\frac{7}{4}x+}\mathrm{\cdots },$$

ⓘ

Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.4 $$\frac{a_1}{1-x+}\frac{-\frac{1}{2}}{\frac{1}{2}(3-x)+}\frac{-\frac{2}{3}}{\frac{1}{3}(5-x)+}\frac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}\mathrm{\cdots },$$

ⓘ

Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

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18.13.5 $$\frac{1}{2x+}\frac{-2}{2x+}\frac{-4}{2x+}\frac{-6}{2x+}\mathrm{\cdots }.$$

ⓘ

Symbols:

$x$: real variable

Permalink:

http://dlmf.nist.gov/18.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.13, §18.13 and Ch.18

…

8: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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17.9.12 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,\frac{de}{abc})=\frac{{(e/b,e/c,cq/a,q/d;q)}_{\mathrm{\infty }}}{{(e,cq/d,q/a,e/(bc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c,d/a,cq/e}{cq/a,bcq/e};q,\frac{bq}{d})-\frac{{(q/d,eq/d,b,c,d/a,de/(bcq),bc{q}^2/(de);q)}_{\mathrm{\infty }}}{{(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{aq/d,bq/d,cq/d}{q^2/d,eq/d};q,\frac{de}{abc}),$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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 and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.13 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,\frac{de}{abc})=\frac{{(e/b,e/c;q)}_{\mathrm{\infty }}}{{(e,e/(bc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{d/a,b,c}{d,bcq/e};q,q)+\frac{{(d/a,b,c,de/(bc);q)}_{\mathrm{\infty }}}{{(d,e,bc/e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{e/b,e/c,de/(abc)}{de/(bc),eq/(bc)};q,q).$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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 and $q$: complex base

Referenced by:

(17.9.3_5)

Permalink:

http://dlmf.nist.gov/17.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.14 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,b,c}{d,e,f};q,q)=\frac{{(e/a,f/a;q)}_n}{{(e,f;q)}_n}{a}^n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,d/b,d/c}{d,a{q}^{1-n}/e,a{q}^{1-n}/f};q,q)=\frac{{(a,ef/(ab),ef/(ac);q)}_n}{{(e,f,ef/(abc);q)}_n}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},e/a,f/a,ef/(abc)}{ef/(ab),ef/(ac),q^{1-n}/a};q,q).$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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 and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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17.9.16 $${}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{a,q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e,f}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^2{q}^2}{bcdef})=\frac{{(aq,aq/(de),aq/(df),aq/(ef);q)}_{\mathrm{\infty }}}{{(aq/d,aq/e,aq/f,aq/(def);q)}_{\mathrm{\infty }}}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{aq/(bc),d,e,f}{aq/b,aq/c,def/a};q,q)+\frac{{(aq,aq/(bc),d,e,f,a^2{q}^2/(bdef),a^2{q}^2/(cdef);q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/e,aq/f,a^2{q}^2/(bcdef),def/(aq);q)}_{\mathrm{\infty }}}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{aq/(de),aq/(df),aq/(ef),a^2{q}^2/(bcdef)}{a^2{q}^2/(bdef),a^2{q}^2/(cdef),a{q}^2/(def)};q,q).$$

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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 and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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17.9.17 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{aq/b,aq/c};q,\frac{aqz}{bc})=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}}{}_5{}^{}\varphi _4^{}(\genfrac{}{}{0pt}{}{a^{\frac{1}{2}},-a^{\frac{1}{2}},{(aq)}^{\frac{1}{2}},-{(aq)}^{\frac{1}{2}},aq/(bc)}{aq/b,aq/c,az,q/z};q,q).$$

ⓘ

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, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

…

9: 4.25 Continued Fractions

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4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

…

10: 10.67 Asymptotic Expansions for Large Argument

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10.67.9 $${\mathrm{ber}}^{2}x+{\mathrm{bei}}^{2}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^2}-\frac{33}{256\sqrt{2}}\frac{1}{x^3}-\frac{1797}{8192}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.27

Permalink:

http://dlmf.nist.gov/10.67.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

►

10.67.10 $$\mathrm{ber}x{\mathrm{bei}}^{\prime }x-{\mathrm{ber}}^{\prime }x\mathrm{bei}x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}+\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}+\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.28

Permalink:

http://dlmf.nist.gov/10.67.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

►

10.67.11 $$\mathrm{ber}x{\mathrm{ber}}^{\prime }x+\mathrm{bei}x{\mathrm{bei}}^{\prime }x\sim \frac{{\mathrm{e}}^{x\sqrt{2}}}{2\pi x}\left(\frac{1}{\sqrt{2}}-\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^2}-\frac{45}{512}\frac{1}{x^3}+\frac{315}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.29

Permalink:

http://dlmf.nist.gov/10.67.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

… ►

10.67.14 $$\ker x{\mathrm{kei}}^{\prime }x-{\ker }^{\prime }x\mathrm{kei}x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}{x}+\frac{9}{64\sqrt{2}}\frac{1}{x^2}-\frac{39}{512}\frac{1}{x^3}+\frac{75}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.32

Permalink:

http://dlmf.nist.gov/10.67.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

►

10.67.15 $$\ker x{\ker }^{\prime }x+\mathrm{kei}x{\mathrm{kei}}^{\prime }x\sim -\frac{\pi }{2x}{\mathrm{e}}^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}{x}-\frac{15}{64\sqrt{2}}\frac{1}{x^2}+\frac{45}{512}\frac{1}{x^3}+\frac{315}{8192\sqrt{2}}\frac{1}{x^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $x$: real variable

A&S Ref:

9.10.33

Permalink:

http://dlmf.nist.gov/10.67.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(ii), §10.67 and Ch.10

…