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1: 31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
2: 31.12 Confluent Forms of Heun’s Equation
Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . … This has irregular singularities at z = 0 and , each of rank 1 . … This has a regular singularity at z = 0 , and an irregular singularity at of rank 2 . … This has one singularity, an irregular singularity of rank 3 at z = . …
3: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with N + 1 regular singularities at z = a j , j = 1 , 2 , , N , and at , is given by
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z - a j ) d w d z + ( j = 1 N q j z - a j ) w = 0 , j = 1 N q j = 0 .
The exponents at the finite singularities a j are { 0 , 1 - γ j } and those at are { α , β } , where …The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N - 2 free accessory parameters q j . …
31.14.3 w ( z ) = ( j = 1 N ( z - a j ) - γ j / 2 ) W ( z ) ,
4: 31.1 Special Notation
Sometimes the parameters are suppressed.
5: 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 . …
6: 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. …
7: 1.10 Functions of a Complex Variable
Then z = z 0 is an isolated singularity of f ( z ) . …Lastly, if a n 0 for infinitely many negative n , then z 0 is an isolated essential singularity. …
8: 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). …
9: 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). …
10: 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