About the Project

irregular solutions

AdvancedHelp

(0.002 seconds)

11—20 of 24 matching pages

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
It has a regular singularity at the origin with indices 1 2 ± μ , and an irregular singularity at infinity of rank one.
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
For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, F ( η , ρ ) and G ( η , ρ ) , or f ( ϵ , ; r ) and h ( ϵ , ; r ) , to determine the scattering S -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). …
14: 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).
§10.2(ii) Standard Solutions
This solution of (10.2.1) is an analytic function of z , except for a branch point at z = 0 when ν is not an integer. … Each solution has a branch point at z = 0 for all ν . …
§10.2(iii) Numerically Satisfactory Pairs of Solutions
15: 16.21 Differential Equation
16.21.1 ( ( 1 ) p m n z ( ϑ a 1 + 1 ) ( ϑ a p + 1 ) ( ϑ b 1 ) ( ϑ b q ) ) w = 0 ,
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 = . … A fundamental set of solutions of (16.21.1) is given by …
16: 10.47 Definitions and Basic Properties
Equations (10.47.1) and (10.47.2) each have a regular singularity at z = 0 with indices n , n 1 , and an irregular singularity at z = of rank 1 ; compare §§2.7(i)2.7(ii). …
§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 J , Y , H , and ν replaced by 𝗃 , 𝗒 , 𝗁 , and n , respectively. For (10.47.2) numerically satisfactory pairs of solutions are 𝗂 n ( 1 ) ( z ) and 𝗄 n ( z ) in the right half of the z -plane, and 𝗂 n ( 1 ) ( z ) and 𝗄 n ( z ) in the left half of the z -plane. …
17: 28.2 Definitions and Basic Properties
This equation has regular singularities at 0 and 1, both with exponents 0 and 1 2 , and an irregular singular point at . …
§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
§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 z , including the turning point z 0 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 u asymptotic approximations of the solutions w can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on u and α ). …
19: 30.2 Differential Equations
30.2.1 d d z ( ( 1 z 2 ) d w d z ) + ( λ + γ 2 ( 1 z 2 ) μ 2 1 z 2 ) w = 0 .
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 ). …
30.2.4 ( ζ 2 γ 2 ) d 2 w d ζ 2 + 2 ζ d w d ζ + ( ζ 2 λ γ 2 γ 2 μ 2 ζ 2 γ 2 ) w = 0 .
20: 16.8 Differential Equations
All other singularities are irregular. … 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 . … When no b j is an integer, and no two b j differ by an integer, a fundamental set of solutions of (16.8.3) is given by … When p = q + 1 , and no two a j differ by an integer, another fundamental set of solutions of (16.8.3) is given by … 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 . …