q-integrals

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1: 5.18 $q$-Gamma and $q$-Beta Functions

… ►For $q$-integrals see §17.2(v).

2: 19.33 Triaxial Ellipsoids

… ►

19.33.5 $$V(\lambda )=Q{R}_F(a^2+\lambda ,b^2+\lambda ,c^2+\lambda ),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $Q$: charge and $V(\lambda )$: potential

Permalink:

http://dlmf.nist.gov/19.33.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(ii), §19.33 and Ch.19

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3: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

… ►and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions: … ►In the case of the symmetric integrals, with the notation of §19.16(i) we have ►

20.9.3 $$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§19.25(v), §20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

4: 17.2 Calculus

… ►

§17.2(v) Integrals

►If $f(x)$ is continuous at $x=0$, then …

5: Bibliography H

… ►

Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

Encodings:

BibTeX

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6: 22.20 Methods of Computation

… ►If either $\tau$ or $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ is given, then we use $k={\theta }_2^2(0,q)/{\theta }_3^2(0,q)$, $k^{\prime }={\theta }_4^2(0,q)/{\theta }_3^2(0,q)$, $K=\frac{1}{2}\pi {\theta }_3^2(0,q)$, and $K^{\prime }=-\mathrm{i}\tau K$, obtaining the values of the theta functions as in §20.14. …

7: 19.21 Connection Formulas

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19.21.15 $$p{R}_J(0,y,z,p)+q{R}_J(0,y,z,q)=3R_F(0,y,z),$$ $pq=yz$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Permalink:

http://dlmf.nist.gov/19.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(iii), §19.21 and Ch.19

8: 18.27 $q$-Hahn Class

… ►In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). …