limiting forms as order tends to integers
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1: 1.13 Differential Equations
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βΊ
§1.13(vii) Closed-Form Solutions
… βΊ§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
… βΊThis is the Sturm-Liouville form of a second order differential equation, where ′ denotes . … βΊTransformation to Liouville normal Form
βΊEquation (1.13.26) with may be transformed to the Liouville normal form …2: 28.2 Definitions and Basic Properties
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βΊThe standard form of Mathieu’s equation with parameters is
…With we obtain the algebraic form of Mathieu’s equation
…With we obtain another algebraic form:
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βΊleads to a Floquet solution.
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βΊ
§28.2(vi) Eigenfunctions
…3: 28.12 Definitions and Basic Properties
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βΊ
§28.12(ii) Eigenfunctions
… βΊHowever, these functions are not the limiting values of as . … βΊAgain, the limiting values of and as are not the functions and defined in §28.2(vi). …4: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
βΊAs with and () fixed, …5: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
… βΊand , are linearly independent solutions of (10.24.1): … βΊIn consequence of (10.24.6), when is large and comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … … βΊ6: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
… βΊand , are real and linearly independent solutions of (10.45.1): … βΊThe corresponding result for is given by … βΊ … βΊ7: 26.3 Lattice Paths: Binomial Coefficients
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βΊ
§26.3(i) Definitions
βΊ is the number of ways of choosing objects from a collection of distinct objects without regard to order. is the number of lattice paths from to . …The number of lattice paths from to , , that stay on or above the line is … βΊ§26.3(v) Limiting Form
…8: 34.8 Approximations for Large Parameters
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βΊFor large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large.
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βΊ
34.8.1
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βΊ
34.8.2
βΊand the symbol denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i).
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9: 29.5 Special Cases and Limiting Forms
§29.5 Special Cases and Limiting Forms
… βΊ
29.5.4
βΊ
29.5.5
even,
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βΊIf and in such a way that (a positive constant), then
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