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classification of singularities

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1: 16.8 Differential Equations
§16.8(i) Classification of Singularities
2: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. …
3: 1.13 Differential Equations
For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
4: 3.7 Ordinary Differential Equations
For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
5: 31.2 Differential Equations
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, { } , can be transformed into (31.2.1). …
6: 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 = . …
7: 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 …The exponents at the finite singularities a j are { 0 , 1 - γ j } and those at are { α , β } , where …
α β = j = 1 N a j q j .
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 ) ,
8: 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 = . …
9: 2.8 Differential Equations with a Parameter
§2.8(i) Classification of Cases
The form of the asymptotic expansion depends on the nature of the transition points in D , that is, points at which f ( z ) has a zero or singularity. …