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1: 30.2 Differential Equations
§30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
2: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
This is the hypergeometric differential equation. … The ( 6 3 ) = 20 connection formulas for the principal branches of Kummer’s solutions are: …
3: 31.2 Differential Equations
§31.2 Differential Equations
§31.2(i) Heun’s Equation
§31.2(ii) Normal Form of Heun’s Equation
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
4: 29.2 Differential Equations
§29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
5: 32.2 Differential Equations
§32.2 Differential Equations
§32.2(i) Introduction
The six Painlevé equations P I P VI  are as follows: …
§32.2(ii) Renormalizations
6: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
7: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 1 ) w ′′ + ζ w + ( 4 q ζ 2 2 q a ) w = 0 , ζ = cosh z .
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ζ 1 / 2 e ± 2 i h ζ as ζ in the respective sectors | ph ( i ζ ) | 3 2 π δ , δ being an arbitrary small positive constant. …
8: 18.42 Software
For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …
9: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • G. Szegő (1933) Asymptotische Entwicklungen der Jacobischen Polynome. Schr. der König. Gelehr. Gesell. Naturwiss. Kl. 10, pp. 33–112.
  • 10: 28 Mathieu Functions and Hill’s Equation
    Chapter 28 Mathieu Functions and Hill’s Equation