integral representation of solutions

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1: 31.10 Integral Equations and Representations
Kernel Functions
For suitable choices of the branches of the $P$-symbols in (31.10.9) and the contour $C$, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …
2: Bibliography V
• H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
• 3: 9.12 Scorer Functions
The general solution is given by …Standard particular solutions are …where …
4: 9.13 Generalized Airy Functions
Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … Connection formulas for the solutions of (9.13.31) include …
5: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions. …
6: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … …
§15.17(iii) Group Representations
By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …
7: 15.19 Methods of Computation
As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. …
§15.19(iii) IntegralRepresentations
The representation (15.6.1) can be used to compute the hypergeometric function in the sector $|\operatorname{ph}\left(1-z\right)|<\pi$. …
8: Bibliography R
• W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.
• W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
• W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.
• W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
• W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.
10: Bibliography S
• B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.
• R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.
• H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
• R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).
• R. Sips (1965) Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.