irregular solutions
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11: 2.9 Difference Equations
§2.9 Difference Equations
… ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). Formal solutions are … ►But there is an independent solution … ►12: 13.14 Definitions and Basic Properties
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►It has a regular singularity at the origin with indices , and an irregular singularity at infinity of rank one.
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Standard Solutions
►Standard solutions are: … ►§13.14(v) Numerically Satisfactory Solutions
►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are …13: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, and , or and , to determine the scattering -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951).
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14: 10.2 Definitions
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►This differential equation has a regular singularity at with indices , and an irregular singularity at of rank ; compare §§2.7(i) and 2.7(ii).
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§10.2(ii) Standard Solutions
… ►This solution of (10.2.1) is an analytic function of , except for a branch point at when is not an integer. … ►Each solution has a branch point at for all . … ►§10.2(iii) Numerically Satisfactory Pairs of Solutions
…15: 16.21 Differential Equation
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16.21.1
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►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at .
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►A fundamental set of solutions of (16.21.1) is given by
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16: 10.47 Definitions and Basic Properties
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►Equations (10.47.1) and (10.47.2) each have a regular singularity at with indices , , and an irregular singularity at of rank ; compare §§2.7(i)–2.7(ii).
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§10.47(ii) Standard Solutions
… ►§10.47(iii) Numerically Satisfactory Pairs of Solutions
►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols , , , and replaced by , , , and , respectively. ►For (10.47.2) numerically satisfactory pairs of solutions are and in the right half of the -plane, and and in the left half of the -plane. …17: 28.2 Definitions and Basic Properties
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►This equation has regular singularities at 0 and 1, both with exponents 0 and , and an irregular singular point at .
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§28.2(iv) Floquet Solutions
… ►The Fourier series of a Floquet solution …leads to a Floquet solution. … ►For the connection with the basic solutions in §28.2(ii), …18: 10.72 Mathematical Applications
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§10.72(i) Differential Equations with Turning Points
►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. … ►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. … ►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 ). …19: 30.2 Differential Equations
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30.2.1
►This equation has regular singularities at with exponents and an irregular singularity of rank 1 at (if ).
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30.2.4
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20: 16.8 Differential Equations
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►All other singularities are irregular.
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►Equation (16.8.4) has a regular singularity at , and an irregular singularity at , whereas (16.8.5) has regular singularities at , , and .
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►When no is an integer, and no two differ by an integer, a fundamental set of solutions of (16.8.3) is given by
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►When , and no two differ by an integer, another fundamental set of solutions of (16.8.3) is given by
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►Thus in the case the regular singularities of the function on the left-hand side at and coalesce into an irregular singularity at .
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