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1: 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 a regular singularity at z = 0 , and an irregular singularity at of rank 2 . …
2: 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 . …
3: 2.7 Differential Equations
§2.7(i) Regular Singularities: Fuchs–Frobenius Theory
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … In a punctured neighborhood N of a regular singularity z 0 Thus a regular singularity has rank 0. … The transformed differential equation either has a regular singularity at t = , or its characteristic equation has unequal roots. …
4: 16.8 Differential Equations
If z 0 is not an ordinary point but ( z - z 0 ) n - j f j ( z ) , j = 0 , 1 , , n - 1 , are analytic at z = z 0 , then z 0 is a regular singularity. … Equation (16.8.4) has a regular singularity at z = 0 , and an irregular singularity at z = , whereas (16.8.5) has regular singularities at z = 0 , 1 , and . … Thus in the case p = q the regular singularities of the function on the left-hand side at α and coalesce into an irregular singularity at . …
5: 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 ). …
6: 15.11 Riemann’s Differential Equation
The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
7: 31.2 Differential Equations
This equation has regular singularities at 0 , 1 , a , , with corresponding exponents { 0 , 1 - γ } , { 0 , 1 - δ } , { 0 , 1 - ϵ } , { α , β } , respectively (§2.7(i)). 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). …
8: 10.2 Definitions
This differential equation has a regular singularity at z = 0 with indices ± ν , and an irregular singularity at z = of rank 1 ; compare §§2.7(i) and 2.7(ii). …
9: 29.2 Differential Equations
This equation has regular singularities at the points 2 p K + ( 2 q + 1 ) i K , where p , q , and K , K are the complete elliptic integrals of the first kind with moduli k , k ( = ( 1 - k 2 ) 1 / 2 ) , respectively; see §19.2(ii). …
10: 33.14 Definitions and Basic Properties
§33.14(i) Coulomb Wave Equation
Again, there is a regular singularity at r = 0 with indices + 1 and - , and an irregular singularity of rank 1 at r = . …