Access
You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen Reader
This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.Asymptotic Expansions for the Coefficient Functions that Arise in Turning Point Problems
W. G. C. Boyd
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 410, No. 1838 (Mar. 9, 1987), pp. 3560
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/2398279
Page Count: 26
 Item Type
 Article
 Thumbnails
 References
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Abstract
We study the uniform asymptotic expansion for a large parameter u of solutions of secondorder linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form $Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^n_{s=0} \frac{A_s(\zeta)}{u^{2s}} + u^{2} \frac{d}{d\zeta} Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^{n1}_{s=0} \frac{B_s(\zeta)}{u^{2s}} + \epsilon^{(\alpha)}_n (u,\zeta)$ for α = 0, 1, 2, with bounds on ε(α) n. We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai(u2/3ζ) A (u, ζ) + u2(d/dζ) Ai (u2/3ζ) B (u, ζ), where Ai denotes any solution of Airy's equation. The coefficient functions A (u, ζ) and B (u, ζ) are the focus of our attention: we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u2, with explicit error bounds. We apply our theory to Bessel functions.
Page Thumbnails

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences © 1987 Royal Society