The asymptotic expansion of integral functions defined by Taylor series

Abstract In a former paper I deduced the asymptotic expansion of the integral function f(x) “ 0(n) xn !=0 rfn+fl) &{{k)> 0) for large x from asymptotic properties of the function (f>{t). In particular, had to be regular and its asymptotic behaviour had to satisfy a certain ‘condition A ’ throughout the half-plane R{kt) > K. My results included as special cases most of the known results about the asymptotic expansion of integral functions. In the present paper the class of functions is widened and the previous theorems completed. I also show that my results are now, in an obvious sense, best possible; that is, if the conditions stated in my theorems are further relaxed, the conclusions are false.

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