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11: 16.21 Differential Equation
With the classification of §16.8(i), when p < q the only singularities of (16.21.1) are a regular singularity at z = 0 and an irregular singularity at z = . When p = q the only singularities of (16.21.1) are regular singularities at z = 0 , ( - 1 ) p - m - n , and . …
12: 36.14 Other Physical Applications
These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge. Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. …
13: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
14: 31.18 Methods of Computation
Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …
15: 16.8 Differential Equations
§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
16: Notices
The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …
17: 30.2 Differential Equations
This equation has regular singularities at z = ± 1 with exponents ± 1 2 μ and an irregular singularity of rank 1 at z = (if γ 0 ). … …
18: Mark J. Ablowitz
ODEs which do not have moveable branch point singularities. …
19: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . …
20: 31.15 Stieltjes Polynomials
31.15.2 j = 1 N γ j / 2 z k - a j + j = 1 j k n 1 z k - z j = 0 , k = 1 , 2 , , n .
31.15.6 a j < a j + 1 , j = 1 , 2 , , N - 1 ,
If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index m = ( m 1 , m 2 , , m N - 1 ) , where each m j is a nonnegative integer, there is a unique Stieltjes polynomial with m j zeros in the open interval ( a j , a j + 1 ) for each j = 1 , 2 , , N - 1 . …