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11: 32.7 Bäcklund Transformations

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§32.7(vii) Sixth Painlevé Equation

… ►The transformations ${𝒮}_j$, for $j=1,2,3$, generate a group of order 24. … ►for $j=0,1,2,\mathrm{\infty }$, where … ►The quartic transformation …transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta =\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_2,{\beta }_2,{\gamma }_2,{\delta }_2)=(16\alpha ,0,0,\frac{1}{2})$. …

12: Bibliography Q

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A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

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A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

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External Links:

ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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…

13: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►For $s=0,1,2$, … ►Similarly for $U^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$. … ►

§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

… ►Similarly for $U^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$. …

14: 8.17 Incomplete Beta Functions

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8.17.5 $$I_x(m,n-m+1)=\sum _{j=m}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)x^j{(1-x)}^{n-j},$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

6.6.4 (Relation to the Binomial Expansion)

Referenced by:

§8.17(i), §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E5

Encodings:

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See also:

Annotations for §8.17(i), §8.17 and Ch.8

… ►The $4m$ and $4m+1$ convergents are less than $I_x(a,b)$, and the $4m+2$ and $4m+3$ convergents are greater than $I_x(a,b)$. … ►The expansion (8.17.22) converges rapidly for . For $x>(a+1)/(a+b+2)$ or , more rapid convergence is obtained by computing $I_{1-x}(b,a)$ and using (8.17.4). … ►

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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15: 11.10 Anger–Weber Functions

… ►For $n=1,2,3,\mathrm{\dots }$, … ►

§11.10(vii) Special Values

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11.10.26 ${𝐄}_0\left(z\right)=-{𝐇}_0\left(z\right),$ ${𝐄}_1\left(z\right)={\displaystyle \frac{2}{\pi }}-{𝐇}_1\left(z\right).$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E26

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

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See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.28 $${\frac{\partial }{\partial \nu }{𝐄}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi J_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E28

Encodings:

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See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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16: 28.4 Fourier Series

… ►For $n=0,1,2,3,\mathrm{\dots }$, … ►For fixed $s=1,2,3,\mathrm{\dots }$ and fixed $m=1,2,3,\mathrm{\dots }$, … ►

§28.4(vii) Asymptotic Forms for Large $m$

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28.4.25 $$\frac{A_{2m+1}^{2n+1}(q)}{A_1^{2n+1}(q)}=\frac{{(-1)}^{m+1}}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}{\left(\frac{q}{4}\right)}^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}^{\prime }(\frac{1}{2}\pi ;a_{2n+1}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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28.4.27 $$\frac{B_{2m}^{2n+2}(q)}{B_2^{2n+2}(q)}=\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{q}{4}\right)}^m\frac{q\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{I}}^{\prime }(\frac{1}{2}\pi ;b_{2n+2}\left(q\right),q)}.$$

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Symbols:

$O\left(x\right)$: order not exceeding, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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17: 13.2 Definitions and Basic Properties

… ►Similarly, when $a-b+1=-n$, $n=0,1,2,\mathrm{\dots }$, … ►When $b=n+1$, $n=0,1,2,\mathrm{\dots }$, and $a\ne 0,-1,-2,\mathrm{\dots }$, … ►When $b=n+1$, $n=0,1,2,\mathrm{\dots }$, and $a=-m$, $m=0,1,2,\mathrm{\dots }$, … ►if $a=1,2,\mathrm{\dots },n$. … ►

§13.2(vii) Connection Formulas

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18: 1.15 Summability Methods

… ►uniformly for $\theta \in [\delta ,2\pi -\delta ]$. … ►For $n=0,1,2,\mathrm{\dots }$, … ► $n=0,1,2,\mathrm{\dots }$, where … ►

§1.15(vii) Fractional Derivatives

… ►Note that ${𝐷}^{1/2}𝐷\ne {𝐷}^{3/2}$. …

19: Bibliography E

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Angular Momentum in Quantum Mechanics. 3rd printing, with corrections, 2nd edition, Princeton University Press, Princeton, NJ.

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Notes:

Reprinted in 1996.

External Links:

ISBN 0-691-02589-4, MathReview, ZentralBlatt

Referenced by:

§14.30(i), §14.30(ii), §14.30(iii), §14.30(iv), §14.30(iv), §14.31(iii), (34.1.1), §34.1, §34.1, §34.13, §34.2, §34.3(i), §34.3(i), §34.3(ii), §34.3(iii), §34.3(iv), §34.3(vii), §34.3(vii), §34.4, §34.5(i), §34.5(ii), §34.5(iii), §34.5(iv), §34.5(vi), §34.6, §34.7(i), §34.7(ii), §34.7(iv), §34.7(vi), Ch.34, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.

Encodings:

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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External Links:

ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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The asymptotic behaviour of the inhomogeneous Airy function $\mathrm{Hi}(z)$ . Math. Chronicle 12, pp. 99–104.

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External Links:

ISSN 0581-1155, MathReview (Ram Kishore Saxena), ZentralBlatt

Referenced by:

(9.12.24), §9.12(vii).

Encodings:

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20: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►with $a_n\ne 0$, $\forall n$, can be computed recursively for $n=2,3,\mathrm{\dots }$. … ►

Example 2. Weber Function

… ►for $n=1,2,\mathrm{\dots }$, and as $n\to \mathrm{\infty }$ … ►

§3.6(vii) Linear Difference Equations of Other Orders

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