equivalent equation for contiguous functions
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1: 28.2 Definitions and Basic Properties
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28.2.1
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►Equivalently,
…If or , or equivalently, , then is a double root of the characteristic equation, otherwise it is a simple root.
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►An equivalent formulation is given by
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§28.2(vi) Eigenfunctions
…2: 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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►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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§28.20(iv) Radial Mathieu Functions ,
…3: 30.2 Differential Equations
§30.2 Differential Equations
►§30.2(i) Spheroidal Differential Equation
… ► … ►The Liouville normal form of equation (30.2.1) is … ►If , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).4: 31.2 Differential Equations
§31.2 Differential Equations
►§31.2(i) Heun’s Equation
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31.2.1
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§31.2(v) Heun’s Equation Automorphisms
… ►Composite Transformations
…5: 29.2 Differential Equations
§29.2 Differential Equations
►§29.2(i) Lamé’s Equation
… ►§29.2(ii) Other Forms
… ►we have …For the Weierstrass function see §23.2(ii). …6: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
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15.10.1
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Singularity
… ►Singularity
… ►Singularity
…7: 32.2 Differential Equations
§32.2 Differential Equations
… ►The six Painlevé equations – are as follows: … ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of –. … ►However, for special values of the parameters, equations – have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. … ► …8: 31.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main functions treated in this chapter are , , , and the polynomial .
…Sometimes the parameters are suppressed.
, | real variables. |
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