TY - JOUR
T1 - Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions <em>p̅s̅<sup>r</sup><sub>n</sub></em>(<em>η</em>,<em>h</em>) and <em>q̅s̅<sup>r</sup><sub>n</sub></em>(<em>η</em>,<em>h</em>) for large <em>h</em>
JF - Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
JO - Proc R Soc Lond A Math Phys Sci
SP - 545
LP - 555
M3 - 10.1098/rspa.1971.0048
VL - 321
IS - 1547
AU -
AU -
Y1 - 1971/03/09
UR - http://rspa.royalsocietypublishing.org/content/321/1547/545.abstract
N2 - The formal techniques of earlier papers (Jorna 1964a, b, 1965a, b) are applied to the differential equation for oblate spheroidal wavefunctions, y(z, h) say, with h2 large. The integro- differential equation arising in the reformulated Liouville–Green method is solved by: (i) direct iteration, yielding asymptotic expansions valid in the region │z│≃ ½π; (ii) taking its Mellin transform and solving the resulting difference equation iteratively. This approach leads to new asymptotic expansions valid for z ≃ 0 and π, and also to the more general uniform expansion. Both methods yield, concurrently, expansions for the eigenvalues and the corresponding functions themselves. As a particular application, expansions are derived for the periodic angular oblate spheroidal wavefunctions p̅s̅(z,h).
ER -