The fundamental purpose of this thesis is to consider the equation y' ' (x)+(A+exn)y(x)=0 , x £ (0,oo) with y'(0) +hy(0)=0 , h positive constant and y(x) has controlling behaviour eip(x) as x —♦ + 0 0 for some positive function p(x). The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im(A) , which was show to be exponentially small like 0(exp[-l/e]) by Paris and Wood. The extension to n = 2 by Brazel, 1/2, Lawless and Wood obtained Im(A) = 0(exp[-l/e ]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available and completely different methods must be employed to obtain the result Im(A) = 0(exp [-1/e 1/n)) . In this thesis we obtain this result by matched asymptotics across the turning pointnearest to the positive real axis by WKB type approximate solution which includes the earlier result of n = 1,2.