ABSTRACT

When x lies off the real axis (8.01) diverges. To obtain the analytic continuation of Ai(x) into the complex plane we transform this integral into a contour integral, as follows. Set t = v/i. Then

SW Ai (x) = - cosh (3v3 -xu) dv = - exp (3v3 -xu) do. ni o Assume temporarily that x is positive and consider

I ( R ) = ~ ~ lexp(3v3-xv)dvl,

where R is a large positive number, and the integration path is the shorter arc of the circle Ivl= R. Substituting v = iRe-'@I3 and applying Jordan's inequality (3.13) we derive

Hence I(R) vanishes as R -, a. Clearly the same is true of the corresponding integral along the conjugate path.