11—20 of 82 matching pages
… ►This equation has regular singularities at with exponents and an irregular singularity of rank 1 at (if ). … …
12: Mark J. Ablowitz
… ►ODEs which do not have moveable branch point singularities. …
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials… ►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities . …
31.15.2 .… ►
31.15.6 ,… ►
31.15.7 .… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index , where each is a nonnegative integer, there is a unique Stieltjes polynomial with zeros in the open interval for each . …
§2.7(i) Regular Singularities: Fuchs–Frobenius Theory… ►Other points are singularities of the differential equation. …All other singularities are classified as irregular. … ►
§2.7(ii) Irregular Singularities of Rank 1… ►Thus a regular singularity has rank 0. …
§15.11(i) Equations with Three Singularities►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). … ►Cases in which there are fewer than three singularities are included automatically by allowing the choice for exponent pairs. … ►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 …
… ►These expansions are uniform with respect to , including the turning point 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. … ►These asymptotic expansions are uniform with respect to , including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
18: Bibliography O
Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one.
Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity.
Methods Appl. Anal. 4 (4), pp. 375–403.