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21: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

… ►

17.7.5 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{e,f};q,q)+\frac{{(q/e,a,b,c,qf/e;q)}_{\mathrm{\infty }}}{{(e/q,aq/e,bq/e,cq/e,f;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{aq/e,bq/e,cq/e}{q^2/e,qf/e};q,q)=\frac{{(q/e,f/a,f/b,f/c;q)}_{\mathrm{\infty }}}{{(aq/e,bq/e,cq/e,f;q)}_{\mathrm{\infty }}},$$

ⓘ

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/17.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

… ►where $\lambda =-c{(ab/q)}^{\frac{1}{2}}$. … ►

17.7.14 $${}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{a,q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e,q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,a{q}^{n+1}};q,q)=\frac{{(aq,aq/(bc),aq/(bd),aq/(cd);q)}_n}{{(aq/b,aq/c,aq/d,aq/(bcd);q)}_n},$$

ⓘ

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

Referenced by:

§17.7(iii)

Permalink:

http://dlmf.nist.gov/17.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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17.7.15 $${}_6{}^{}\varphi _5^{}(\genfrac{}{}{0pt}{}{a,q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{aq}{bcd})=\frac{{(aq,aq/(bc),aq/(bd),aq/(cd);q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/(bcd);q)}_{\mathrm{\infty }}},$$

ⓘ

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/17.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

… ►

17.7.17 $${}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{a,q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e,f}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,q)-\frac{b}{a}\frac{{(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f;q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b^{2}q/a;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{b^2/a,qb{a}^{-\frac{1}{2}},-qb{a}^{-\frac{1}{2}},b,bc/a,bd/a,be/a,bf/a}{b{a}^{-\frac{1}{2}},-b{a}^{-\frac{1}{2}},bq/a,bq/c,bq/d,bq/e,bq/f};q,q)=\frac{{(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q)}_{\mathrm{\infty }}}{{(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q)}_{\mathrm{\infty }}},$$

ⓘ

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/17.7.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

…

22: 22.13 Derivatives and Differential Equations

… ►(The modulus $k$ is suppressed throughout the table.) … ►

22.13.2 $${\left(\frac{d}{dz}\mathrm{cn}(z,k)\right)}^2=\left(1-{\mathrm{cn}}^2(z,k)\right)\left(k_{}^{\prime }{}_{}{}^2+k^2{\mathrm{cn}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

►

22.13.3 $${\left(\frac{d}{dz}\mathrm{dn}(z,k)\right)}^2=\left(1-{\mathrm{dn}}^2(z,k)\right)\left({\mathrm{dn}}^2(z,k)-k_{}^{\prime }{}_{}{}^2\right).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.5 $${\left(\frac{d}{dz}\mathrm{sd}(z,k)\right)}^2=\left(1-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k)\right)\left(1+k^2{\mathrm{sd}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

►

22.13.6 $${\left(\frac{d}{dz}\mathrm{nd}(z,k)\right)}^2=\left({\mathrm{nd}}^2(z,k)-1\right)\left(1-k_{}^{\prime }{}_{}{}^2{\mathrm{nd}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

…

23: 9.10 Integrals

… ►

9.10.4 $${\int }_x^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt\sim \frac{1}{2}{\pi }^{-1/2}{x}^{-3/4}\exp \left(-\frac{2}{3}{x}^{3/2}\right),$$ $x\to \mathrm{\infty }$,

ⓘ

Symbols:

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

Proof sketch:

Integrate the leading term of (9.7.5).

Referenced by:

§9.10(ii)

Permalink:

http://dlmf.nist.gov/9.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(ii), §9.10 and Ch.9

… ►

9.10.6 $${\int }_{-\mathrm{\infty }}^x\mathrm{Ai}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\cos \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$ $x\to -\mathrm{\infty }$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Proof sketch:

Integrate the leading terms of (9.7.5) and use (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(ii), §9.10 and Ch.9

►

9.10.7 $${\int }_{-\mathrm{\infty }}^x\mathrm{Bi}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\sin \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$ $x\to -\mathrm{\infty }$.

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

Proof sketch:

Integrate the leading terms of (9.7.7) and use (9.10.12)

Referenced by:

§9.10(ii)

Permalink:

http://dlmf.nist.gov/9.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(ii), §9.10 and Ch.9

… ►

9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt,$$ .

ⓘ

Symbols:

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

Proof sketch:

To prove this equation, combine (13.4.5) with (13.6.11) and use (5.5.3).

Referenced by:

(9.10.19), §9.17(iii), §9.5(ii), Erratum (V1.0.10) for Equation (9.10.18)

Permalink:

http://dlmf.nist.gov/9.10.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Ai}\left(z\right)=\frac{z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{2^{7/2}\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

►

9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$ $x>0$,

ⓘ

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 $x$ , $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

Proof sketch:

To prove this equation, combine (9.2.10) with (9.10.18).

Referenced by:

§9.5(i), Erratum (V1.0.10) for Equation (9.10.19)

Permalink:

http://dlmf.nist.gov/9.10.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

…

24: 19.9 Inequalities

… ►

19.9.4 $${\left(\frac{1+k_{}^{\prime }{}_{}{}^{3/2}}{2}\right)}^{2/3}\le \frac{2}{\pi }E\left(k\right)\le {\left(\frac{1+k_{}^{\prime }{}_{}{}^2}{2}\right)}^{1/2}$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.24(i), §19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

… ►The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\le k^2\le 0.9960$. … ►Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi (a+b))$: … ►Throughout this subsection we assume that , $0\le \varphi \le \pi /2$, and $\mathrm{\Delta }=\sqrt{1-k^2{\sin }^2\varphi }>0$. … ►

$L=(1/\sigma )\mathrm{arctanh}\left(\sigma \sin \varphi \right),$ $\sigma =\sqrt{(1+k^2)/2}$,

…

25: 17.13 Integrals

… ►

17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

… ►

17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

… ►

17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

…

26: 7.1 Special Notation

… ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. …

27: 36.9 Integral Identities

… ►

36.9.1 $${|{\mathrm{\Psi }}_1\left(x\right)|}^2=2^{5/3}{\int }_0^{\mathrm{\infty }}{\mathrm{\Psi }}_1\left(2^{2/3}(3u^2+x)\right)du;$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.2 $${(\mathrm{Ai}\left(x\right))}^2=\frac{2^{2/3}}{\pi }{\int }_0^{\mathrm{\infty }}\mathrm{Ai}\left(2^{2/3}(u^2+x)\right)du.$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $x$: real parameter

Referenced by:

§9.5(i)

Permalink:

http://dlmf.nist.gov/36.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.4 $${|{\mathrm{\Psi }}_2\left(x,y\right)|}^2={\int }_0^{\mathrm{\infty }}\left({\mathrm{\Psi }}_1\left(\frac{4u^3+2uy+x}{u^{1/3}}\right)+{\mathrm{\Psi }}_1\left(\frac{4u^3+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

… ►

36.9.6 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=2^{4/5}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}_3\left(2^{4/5}(x+2uy+3u^{2}z+5u^4),0,2^{2/5}(z+10u^2)\right)du.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\int$: integral, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

►

36.9.7 $${|{\mathrm{\Psi }}_3\left(x,y,z\right)|}^2=\frac{2^{7/4}}{5^{1/4}}{\int }_0^{\mathrm{\infty }}\mathrm{\Re }\left({\mathrm{e}}^{2\mathrm{i}u(u^4+z{u}^2+x)}{\mathrm{\Psi }}_2\left(\frac{2^{7/4}}{5^{1/4}}y{u}^{3/4},\sqrt{\frac{2u}{5}}(3z+10u^2)\right)\right)\frac{du}{u^{1/4}}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

…

28: 19.7 Connection Formulas

… ►

$E\left(\mathrm{i}k/k^{\prime }\right)=(1/k^{\prime })E\left(k\right),$

►

$E\left(-\mathrm{i}{k}^{\prime }/k\right)=(1/k)E\left(k^{\prime }\right).$

… ►

§19.7(ii) Change of Modulus and Amplitude

… ►

Reciprocal-Modulus Transformation

… ►

Imaginary-Modulus Transformation

…

29: 22.19 Physical Applications

… ►The subsequent time evolution is always oscillatory with period $4K\left(k\right)/\sqrt{1+2\eta }$ and modulus $k=1/\sqrt{2+{\eta }^{-1}}$: … ►There is bounded oscillatory motion near $x=0$, with period $4K\left(k\right)/\sqrt{1-\eta }$, and modulus $k=1/\sqrt{{\eta }^{-1}-1}$, for initial displacements with $|a|\le \sqrt{1/\beta }$. …As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►Such oscillations, of period $2K\left(k\right)/\sqrt{\eta }$, with modulus $k=1/\sqrt{2-{\eta }^{-1}}$ are given by: …with period $4K\left(k\right)/\sqrt{2\eta -1}$ and modulus $k=1/\sqrt{2-{\eta }^{-1}}$. …

30: 23.17 Elementary Properties

… ►

$\lambda \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)={\mathrm{e}}^{\pi \mathrm{i}/3},$

… ►

$J\left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)=0,$

►

$\eta \left(\mathrm{i}\right)={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)}{2{\pi }^{3/4}}},$

►

$\eta \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)={\displaystyle \frac{3^{1/8}{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^{3/2}}{2\pi }}{\mathrm{e}}^{\pi \mathrm{i}/24}.$

… ►with $q^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.