DLMF: Untitled Document

§21.9 Integrable Equations

►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►

§15.11 Riemann’s Differential Equation

►

§15.11(i) Equations with Three Singularities

… ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$ -symbol: … ►

§15.11(ii) Transformation Formulas

… ►for arbitrary

\lambda

and

\mu

.

… ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

… ►

31.11.2

P_{j}=P\begin{Bmatrix}0&1&\infty&\\ 0&0&\lambda+j&z\\ 1-\gamma&1-\delta&\mu-j&\end{Bmatrix},

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Defines:: $P_{j}$ : General solution of generalized hypergeometric differential equation (locally)
Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer, $\lambda$ and $\mu$
Referenced by:: §31.11(iii)
Permalink:: http://dlmf.nist.gov/31.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

…

… ►

C. L. Tretkoff and M. D. Tretkoff (1984) Combinatorial Group Theory, Riemann Surfaces and Differential Equations. In Contributions to Group Theory, Contemp. Math., Vol. 33, pp. 467–519.

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External Links:: MathReview (William Harvey), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

…

… ►

J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:: ISBN 0-8176-3837-7, MathReview, ZentralBlatt
Referenced by:: §15.17(v).
Encodings:: BibTeX

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… ►

31.10.9

\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\[1.0pt] 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos}^{2}\theta\\[1.0pt] 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\[1.0pt] 0&0&-\frac{1}{2}+\delta+\sigma&{\cos}^{2}\phi\\[1.0pt] 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},

… ►

31.10.11

\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\left(\ifrac{zt}{a}\right)^% {-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-% \sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-\gamma\atop\alpha-\beta+1};% \frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant
Permalink:: http://dlmf.nist.gov/31.10.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.22

\mathcal{K}(r,\theta,\phi)=r^{m}{\sin}^{2p}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos}^{2}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos}^{2}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},

…

… ►

A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:: ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).
Encodings:: BibTeX

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A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

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A. B. Olde Daalhuis and F. W. J. Olver (1998) On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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External Links:: ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt
Referenced by:: §16.25, §2.7(ii), §3.7(iii).
Encodings:: BibTeX

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F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:: FullText, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

… ►

P. J. Olver (1993b) Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.

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External Links:: ISBN 0-387-94007-3; 0-387-95000-1, MathReview, ZentralBlatt
Referenced by:: §32.13(i).
Encodings:: BibTeX

…

… ►

5.9.11

\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-\frac{1}{2\pi i}\int_{-c-% \infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(\pi s\right)}\zeta\left(-s% \right)\,\mathrm{d}s,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $s$ : real or complex variable and $z$ : complex variable
Referenced by:: §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
Permalink:: http://dlmf.nist.gov/5.9.E11
Encodings:: pMML, png, TeX
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z+1\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z+1\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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… ►

Riemann Integrals

… ►If the limit exists then

f

is called Riemann integrable. … ►A generalization of the Riemann integral is the Stieltjes integral

\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)

, where

\alpha(x)

is a nondecreasing function on the closure of

(a,b)

, which may be bounded, or unbounded, and

\,\mathrm{d}\alpha(x)

is the Stieltjes measure. …Stieltjes integrability for

f

with respect to

\alpha

can be defined similarly as Riemann integrability in the case that

\alpha(x)

is differentiable with respect to

x

; a generalization follows below. … ►For

\alpha(x)

nondecreasing on the closure

I

of an interval

(a,b)

, the measure

\,\mathrm{d}\alpha

is absolutely continuous if

\alpha(x)

is continuous and there exists a weight function

w(x)\geq 0

, Riemann (or Lebesgue) integrable on finite subintervals of

I

, such that …

Riemann differential equation

1—10 of 34 matching pages

1: 21.9 Integrable Equations

§21.9 Integrable Equations

2: 15.11 Riemann’s Differential Equation

§15.11 Riemann’s Differential Equation

§15.11(i) Equations with Three Singularities

§15.11(ii) Transformation Formulas

3: 15.17 Mathematical Applications

4: 31.11 Expansions in Series of Hypergeometric Functions

5: Bibliography T

6: Bibliography G

7: 31.10 Integral Equations and Representations

8: Bibliography O

9: 5.9 Integral Representations

10: 1.4 Calculus of One Variable

Riemann Integrals