second solutions
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31—40 of 102 matching pages
31: 31.14 General Fuchsian Equation
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►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions).
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32: 14.3 Definitions and Hypergeometric Representations
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►The following are real-valued solutions of (14.2.2) when , and .
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Ferrers Function of the Second Kind
… ►The following are solutions of (14.2.2) when , and . … ►Associated Legendre Function of the Second Kind
… ►As standard solutions of (14.2.2) we take the pair and , where …33: 10.45 Functions of Imaginary Order
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►and , are real and linearly independent solutions of (10.45.1):
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►In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or .
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34: 1.13 Differential Equations
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►For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977).
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35: 14.20 Conical (or Mehler) Functions
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►Lastly, for the range , is a real-valued solution of (14.20.1); in terms of (which are complex-valued in general):
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36: 10.27 Connection Formulas
37: 14.21 Definitions and Basic Properties
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►Standard solutions: the associated Legendre functions , , , and .
…When is complex , , and are defined by (14.3.6)–(14.3.10) with replaced by : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when , and by continuity elsewhere in the -plane with a cut along the interval ; compare §4.2(i).
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§14.21(ii) Numerically Satisfactory Solutions
►When and , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane is given by and . …38: 29.10 Lamé Functions with Imaginary Periods
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29.10.3
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►are solutions of (29.2.1).
The first and the fourth functions have period ; the second and the third have period .
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39: 32.6 Hamiltonian Structure
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►are solutions of (32.6.3) and (32.6.4).
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