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11: 23.22 Methods of Computation

… ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

ⓘ

Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

…

12: DLMF Project News

error generating summary

13: 5.2 Definitions

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5.2.5 $${\left(a\right)}_n=\mathrm{\Gamma }\left(a+n\right)/\mathrm{\Gamma }\left(a\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

A&S Ref:

6.1.22

Referenced by:

(25.11.28), (25.5.7), (25.8.2)

Permalink:

http://dlmf.nist.gov/5.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

►

5.2.6 $${\left(-a\right)}_n={(-1)}^n{\left(a-n+1\right)}_n,$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.2(iii), Erratum (V1.0.17) for Subsection 5.2(iii)

Permalink:

http://dlmf.nist.gov/5.2.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

►

5.2.7

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation was added.

See also:

Annotations for §5.2(iii), §5.2 and Ch.5

…

14: 6.1 Special Notation

… ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …

15: 28.8 Asymptotic Expansions for Large $q$

… ►They are derived by rigorous analysis and accompanied by strict and realistic error bounds. …

16: 17.2 Calculus

… ►For $n=0,1,2,\mathrm{\dots }$, ►

17.2.1 $${(a;q)}_n=(1-a)(1-aq)\mathrm{\cdots }(1-a{q}^{n-1}),$$

ⓘ

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.E1

Encodings:

pMML, png, TeX

See also:

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

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17.2.3 $${(a;q)}_{\nu }=\prod _{j=0}^{\mathrm{\infty }}\left(\frac{1-a{q}^j}{1-a{q}^{\nu +j}}\right),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

17.2.4 $${(a;q)}_{\mathrm{\infty }}=\prod _{j=0}^{\mathrm{\infty }}(1-a{q}^j),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

17.2.7 $${(a;q^{-1})}_n={(a^{-1};q)}_n{(-a)}^n{q}^{-\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/17.2.E7

Encodings:

pMML, png, TeX

See also:

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

…

17: 33.2 Definitions and Basic Properties

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33.2.9 $${\theta }_{\mathrm{\ell }}(\eta ,\rho )=\rho -\eta \ln \left(2\rho \right)-\frac{1}{2}\mathrm{\ell }\pi +{\sigma }_{\mathrm{\ell }}\left(\eta \right),$$

ⓘ

Defines:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.5.5

Referenced by:

§33.10(i), §33.11

Permalink:

http://dlmf.nist.gov/33.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

… ►

33.2.10 $${\sigma }_{\mathrm{\ell }}\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right),$$

ⓘ

Defines:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

A&S Ref:

14.5.6

Referenced by:

§33.10(ii), §33.10(iii), §33.13, §33.2(iii), §33.25, §5.20

Permalink:

http://dlmf.nist.gov/33.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

… ► ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift. … ►Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …

18: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

… ►

17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }};$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-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, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)

Permalink:

http://dlmf.nist.gov/17.5.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.11):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.2 $${}_1{}^{}\varphi _0^{}(a;-;q,z)=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}},$$ ;

ⓘ

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/17.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3 $${}_1{}^{}\varphi _0^{}(q^{-n};-;q,z)={(z{q}^{-n};q)}_n.$$

ⓘ

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, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.4 $${}_1{}^{}\varphi _0^{}(0;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(q;q)}_n}=\frac{1}{{(z;q)}_{\mathrm{\infty }}},$$ ;

ⓘ

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, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii)

Permalink:

http://dlmf.nist.gov/17.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.5 $${}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a}{c};q,c/a)=\frac{{(c/a;q)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}}.$$

ⓘ

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 and $q$: complex base

Referenced by:

Erratum (V1.1.10) for Equation (17.5.5)

Permalink:

http://dlmf.nist.gov/17.5.E5

Encodings:

pMML, png, TeX

Correction (effective with 1.1.10):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

19: 17.3 $q$-Elementary and $q$-Special Functions

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17.3.1 $$e_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{x}^n}{{(q;q)}_n}=\frac{1}{{((1-q)x;q)}_{\mathrm{\infty }}},$$

ⓘ

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

►

17.3.2 $$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$

ⓘ

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

… ►

17.3.3 $${\sin }_q\left(x\right)=\frac{1}{2\mathrm{i}}(e_q\left(\mathrm{i}x\right)-e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}},$$

ⓘ

Defines:

${\sin }_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E3

Encodings:

pMML, png, TeX

See also:

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

►

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

ⓘ

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

… ►

17.3.5 $${\cos }_q\left(x\right)=\frac{1}{2}(e_q\left(\mathrm{i}x\right)+e_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}},$$

ⓘ

Defines:

${\cos }_q\left(x\right)$: $q$-cosine function

Symbols:

$\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $e_q\left(x\right)$: $q$-exponential function, $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E5

Encodings:

pMML, png, TeX

See also:

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

…

20: 18.7 Interrelations and Limit Relations

… ►

18.7.1 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right),$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.27

Referenced by:

§18.15(ii), §18.7(i), §18.7(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.2 $$P_n^{(\alpha ,\alpha )}\left(x\right)=\frac{{\left(\alpha +1\right)}_n}{{\left(2\alpha +1\right)}_n}{C}_n^{(\alpha +\frac{1}{2})}\left(x\right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.20

Referenced by:

§18.6(i), §18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.7 $$T_n^{\ast }\left(x\right)=T_n\left(2x-1\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the first kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.14

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.8 $$U_n^{\ast }\left(x\right)=U_n\left(2x-1\right).$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $U_n^{\ast }\left(x\right)$: shifted Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.15

Permalink:

http://dlmf.nist.gov/18.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.10 $$P_n^{\ast }\left(x\right)=P_n\left(2x-1\right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $P_n^{\ast }\left(x\right)$: shifted Legendre polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

…