On a Uniform Treatment of Darboux’s Method
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Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darboux’s method can be used to give an asymptotic expansion for the coefficient of zn in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.
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