Asymptotic Expansions of Integral Functions

The paper falls into three divisions. In the first part, which is very short, Poincare's arithmetic definition of asymptotic expansion is given. The second part emphasises the function theory and the ideas of the theory of functions of a complex variable, and contour integrals are freely used in this part; distinction is made between the different ''orders" of the asymptotic series; and integrals which may be taken to be their "sums" are obtained.