(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ .

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

… ►

(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …

35: 19.33 Triaxial Ellipsoids

… ►and the electric capacity

C=Q/V(0)

is given by … ►A conducting elliptic disk is included as the case

c=0

. … ►Let a homogeneous magnetic ellipsoid with semiaxes

a, b, c

, volume

V=4\pi abc/3

, and susceptibility

\chi

be placed in a previously uniform magnetic field

H

parallel to the principal axis with semiaxis

c

. The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength

H/(1+L_{c}\chi)

, where

L_{c}

is the demagnetizing factor, given in cgs units by … ►where

L_{a}

and

L_{b}

are obtained from

L_{c}

by permutation of

a

b

, and

c

. …

36: 32.10 Special Function Solutions

… ►with

C_{1}

C_{2}

arbitrary constants. … ► … ►with

C_{1}

C_{2}

arbitrary constants. … ►with

C_{1}

C_{2}

arbitrary constants. …

37: 6.13 Zeros

… ►

\operatorname{Ci}\left(x\right)

and

\operatorname{si}\left(x\right)

each have an infinite number of positive real zeros, which are denoted by

c_{k}

s_{k}

, respectively, arranged in ascending order of absolute value for

k=0,1,2,\dots

. Values of

c_{1}

and

c_{2}

to 30D are given by MacLeod (1996b). … ►

6.13.2

c_{k},s_{k}\sim\alpha+\frac{1}{\alpha}-\frac{16}{3}\frac{1}{\alpha^{3}}+\frac{% 1673}{15}\frac{1}{\alpha^{5}}-\frac{5\;07746}{105}\frac{1}{\alpha^{7}}+\cdots,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $c_{k}$ : zeros of $\operatorname{Ci}\left(x\right)$ and $s_{k}$ : zeros of $\operatorname{Si}\left(x\right)$
Referenced by:: §6.13, §6.18(iii)
Permalink:: http://dlmf.nist.gov/6.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.13 and Ch.6

►where

\alpha=k\pi

for

c_{k}

, and

\alpha=(k+\frac{1}{2})\pi

for

s_{k}

. …

38: 15.14 Integrals

… ►

15.14.1

\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\,\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},

\min(\Re a,\Re b)>\Re s>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $s$ : nonnegative integer, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.14
Permalink:: http://dlmf.nist.gov/15.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.14 and Ch.15

►Integrals of the form

\int x^{\alpha}(x+t)^{\beta}F\left(a,b;c;x\right)\,\mathrm{d}x

and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). …

39: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

17.8.4

{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.5

{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.6

{{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

►

17.8.7

{{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}}.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

… ►

17.8.8

{{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),

\left|cq^{2}\right|<\left|b^{2}\right|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Source:: Verma and Jain (1983, (3.9))
Referenced by:: §17.6(i), §17.8, Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.8, §17.8 and Ch.17

…

40: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

17.6.13

{{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{% \left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;% q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{% \infty}},

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

… ►

17.6.23

q\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)+(1-% a)\left(1-\frac{abz}{c}\right){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)=% \left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac{abz}{c}\right){{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right),

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

►

17.6.24

(1-c)(q-c)(abz-c){{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)+z(c-a)(c-b){{% }_{2}\phi_{1}}\left({a,b\atop cq};q,z\right)=(c-1)(c(q-c)+z(ca+cb-ab-abq)){{}_% {2}\phi_{1}}\left({a,b\atop c};q,z\right).

ⓘ

Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

… ►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions

a\to q^{a}

b\to q^{b}

c\to q^{c}

, followed by

\lim_{q\to 1-}

. … ►where

|z|<1

|\operatorname{ph}\left(-z\right)|<\pi

, and the contour of integration separates the poles of

\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\sin\left(\pi\zeta\right)

from those of

1/\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}

, and the infimum of the distances of the poles from the contour is positive. …

.梅西和c罗第一次世界杯『网址:mxsty.cc』.时世界杯赛程-m6q3s2-qmymcyumu.com

31—40 of 394 matching pages

31: 2 Asymptotic Approximations

32: 15.20 Software

33: 19 Elliptic Integrals

34: 4.43 Cubic Equations