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Asymptotic Solutions of Second-Order Linear Differential Equations having Almost Coalescent Turning Points, with an Application to the Incomplete Gamma Function

T. M. Dunster
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 452, No. 1949 (Jun. 8, 1996), pp. 1331-1349
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52919
Page Count: 19
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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 Solutions of Second-Order Linear Differential Equations having Almost Coalescent Turning Points, with an Application to the Incomplete Gamma Function
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Abstract

Uniform asymptotic expansions are derived for solutions of the differential equation d2W/dζ 2 = (u2ζ 2 + β u + ψ (u, ζ ))W, which are uniformly valid for u real and large, β bounded (real or complex), and ζ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The function ψ (u, ζ ) is assumed to be holomorphic in this domain, and is o(u/ln(u)) uniformly as u → ∞ . The approximations involve parabolic cylinder functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary incomplete gamma function Γ (α , α x), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying | arg(x)| <3/2π ; in this domain the error term (explicitly bounded) is asymptotically small as either α → ∞ or x → ∞ .

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