Picard–Fuchs equations

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11: 31.11 Expansions in Series of Hypergeometric Functions

… ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. …The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. …Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►Such series diverge for Fuchs–Frobenius solutions. …

12: 1.10 Functions of a Complex Variable

… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. … ►

Picard’s Theorem

… ►Then the equation …

13: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:

ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

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Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, pp. 37–65.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iii).

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14: 2.7 Differential Equations

§2.7 Differential Equations

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§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

►An ordinary point of the differential equation … ►

§2.7(ii) Irregular Singularities of Rank 1

… ►See §2.11(v) for other examples. …

15: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:

MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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External Links:

ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

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16: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

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Kernel Functions

… ►Fuchs–Frobenius solutions $W_m(z)={\tilde{\kappa }}_m{z}^{-\alpha }H\mathrm{\ell }(1/a,q_m;\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;1/z)$ are represented in terms of Heun functions $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ by (31.10.1) with $W(z)=W_m(z)$, $w(z)=w_m(z)$, and with kernel chosen from … ►leads to the kernel equation … ►

17: 28.18 Integrals and Integral Equations

§28.18 Integrals and Integral Equations

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18: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

19: 29.19 Physical Applications

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§29.19(ii) Lamé Polynomials

►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Hargrave (1978) studies high frequency solutions of the delta wing equation. …Roper (1951) solves the linearized supersonic flow equations. Clarkson (1991) solves nonlinear evolution equations. …

20: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

… ►Equation (32.13.3) also has the similarity reduction … ►

§32.13(ii) Sine-Gordon Equation

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§32.13(iii) Boussinesq Equation

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