Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,>4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.