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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 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 – are as follows: … ►§32.2(ii) Renormalizations
… ► …6: 28.2 Definitions and Basic Properties
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§28.2(i) Mathieu’s Equation
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28.2.1
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►This is the characteristic equation of Mathieu’s equation (28.2.1).
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§28.2(iv) Floquet Solutions
… ► …7: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
►When is replaced by , (28.2.1) becomes the modified Mathieu’s equation: ►
28.20.1
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28.20.2
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►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
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8: 18.42 Software
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►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3).
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9: 22.22 Software
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►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C13).
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10: Bibliography
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Algorithm 724: Program to calculate F-percentiles.
ACM Trans. Math. Software 19 (4), pp. 481–483.
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Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders.
ACM Trans. Math. Software 26 (3), pp. 408–414.
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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function.
ACM Trans. Math. Software 9 (4), pp. 494–502.
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Algorithm 556: Exponential integrals.
ACM Trans. Math. Software 6 (3), pp. 420–428.
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Algorithm 588. Fast Hankel transforms using related and lagged convolutions.
ACM Trans. Math. Software 8 (4), pp. 369–370.
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