for derivatives with respect to order
(0.010 seconds)
21—30 of 93 matching pages
21: Bibliography S
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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem. 46 (1), pp. 231–260.
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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem. 49 (7), pp. 1436–1477.
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22: 36.11 Leading-Order Asymptotics
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36.11.2
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23: 14.20 Conical (or Mehler) Functions
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14.20.1
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24: 22.13 Derivatives and Differential Equations
§22.13 Derivatives and Differential Equations
►§22.13(i) Derivatives
… ►Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … ►§22.13(ii) First-Order Differential Equations
… ►§22.13(iii) Second-Order Differential Equations
…25: 1.13 Differential Equations
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►(More generally in (1.13.5) for th-order differential equations, is the coefficient multiplying the th-order derivative of the solution divided by the coefficient multiplying the th-order derivative of the solution, see Ince (1926, §5.2).)
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and belong to domains and
respectively, the coefficients and are continuous functions of both variables, and for each fixed (fixed ) the two functions are analytic in (in ).
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►Here dots denote differentiations with respect to
, and is the Schwarzian derivative:
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►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984).
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►This is the Sturm-Liouville form of a second order differential equation, where ′ denotes .
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26: 10.72 Mathematical Applications
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►These expansions are uniform with respect to
, including the turning point and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order
, where is the limiting value of as .
These asymptotic expansions are uniform with respect to
, including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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►Then for large asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
These approximations are uniform with respect to both and , including , the cut neighborhood of , and .
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27: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
… ►and , are linearly independent solutions of (10.24.1): … ► … ►In this reference and are denoted respectively by and .28: 16.8 Differential Equations
§16.8 Differential Equations
►§16.8(i) Classification of Singularities
… ►Equation (16.8.3) is of order . In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected. … ►For other values of the , series solutions in powers of (possibly involving also ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. …29: 36.12 Uniform Approximation of Integrals
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36.12.3
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