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The solutions of second order linear ordinary differential equations about a turning point of order two
| Content Provider | Semantic Scholar |
|---|---|
| Author | McKelvey, Robert W. |
| Copyright Year | 1955 |
| Abstract | and po(x), p1(x), and the qj(x)'s are suitable restricted bounded functions. Specifically, formulas are sought which represent solutions of (1.1) in a certain asymptotic sense when lXI > N(2). Existing asymptotic theory of the equation (1.1) is known to depend upon the nature of the vanishing of po(x) and p1(x), these being coefficients of positive powers of X. A zero of po(x) has been called a turning point of the equation, the order of the turning point being the order of the zero. In an interval where po(x) -dO, complete asymptotic expansions of a pair of solutions have long been known; this case was the subject of the classical work of Birkhoff [1 ]. A recent advance, due to R. E. Langer [3 ], has brought the theory of the simple turning point to a similar degree of completion. For higher order turning points, on the other hand, the theory is still fragmentary, being most highly developed for a turning point of order two. In this case expansions have in fact been given [2; 4], but only the leading term of an expansion has been expressed in any simple way. For terms after the first, the known expression are extremely complicated, each successive term a degree more involved than the one before. It is the object of the present paper to obtain expansions, valid in the presence of a second order turning point, whose terms to any order are simply expressed. An algorithm will be given which involves only simple power series |
| Starting Page | 103 |
| Ending Page | 123 |
| Page Count | 21 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9947-1955-0069344-7 |
| Volume Number | 79 |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/1955-079-01/S0002-9947-1955-0069344-7/S0002-9947-1955-0069344-7.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9947-1955-0069344-7 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
