ABSTRACT
From (9.03) it can be seen that the right-hand side of (9.04) is bounded by (la,,J+l)/(z("; hence (9.01) is the asymptotic expansion of f(z) as z - + co in any unbounded region.
This solution is not unique. For example, if we change the definition of v(lz1) by replacing the right-hand side of (9.03) by k 121, where k is any positive constant, then (9.02) again has (9.01) as its asymptotic expansion. The infinite class of functions having (9.01) as asymptotic expansion is called the asymproricsum of this series in R. 9.2 The function (9.02) is somewhat artificial in the sense that it is discontinuous on an infinite set of circles. We shall now construct an analyric function with the desired property. The only restriction is that the range of ph z is bounded.
