singularity parameter
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31—40 of 61 matching pages
31: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
…where is a real or complex variable and is a large real or complex parameter.
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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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►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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►In (10.72.1) assume and depend continuously on a real parameter
, has a simple zero and a double pole , except for a critical value , where .
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32: 2.5 Mellin Transform Methods
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►where denotes the Bessel function (§10.2(ii)), and is a large positive parameter.
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►Then as in (2.5.6) and (2.5.7), with , we obtain
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►Furthermore, can be continued analytically to a meromorphic function on the entire -plane, whose singularities are simple poles at , , with principal part
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►The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987).
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§2.5(iii) Laplace Transforms with Small Parameters
…33: 31.7 Relations to Other Functions
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31.7.1
►Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where .
Below are three such reductions with three and two parameters.
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31.7.2
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►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities
, , and , where and are related to as in §19.2(ii).
34: 25.11 Hurwitz Zeta Function
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25.11.30
, ,
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35: 8.12 Uniform Asymptotic Expansions for Large Parameter
§8.12 Uniform Asymptotic Expansions for Large Parameter
… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at , and the Maclaurin series expansion of is given by … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at is given by … ►Inverse Function
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8.12.21
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36: 16.8 Differential Equations
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§16.8(i) Classification of Singularities
… ►All other singularities are irregular. … … ►In each case there are no other singularities. … ►§16.8(iii) Confluence of Singularities
…37: 20.13 Physical Applications
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20.13.1
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►For , with real, (20.13.1) takes the form of a real-time diffusion equation
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20.13.2
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20.13.3
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►In the singular limit , the functions , , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195).
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38: Bibliography W
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Asymptotische Darstellungen der hypergeometrischen Funktion für große Parameter unterschiedlicher Größenordnung.
Z. Anal. Anwendungen 5 (3), pp. 265–276 (German).
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Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter.
Z. Anal. Anwendungen 9 (4), pp. 351–360 (German).
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The analyticity of Jacobian functions with respect to the parameter
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Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
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Asymptotic expansions of Fourier transforms of functions with logarithmic singularities.
J. Math. Anal. Appl. 64 (1), pp. 173–180.
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Asymptotic expansions of Hankel transforms of functions with logarithmic singularities.
Comput. Math. Appl. 3 (4), pp. 271–286.
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39: 1.13 Differential Equations
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