ABSTRACT

Hence by Weierstrass' M-test the integral (1.01) converges uniformly with respect to z in this strip. That r (z ) is holomorphic in the half-plane Rez > Ois a consequence of this result and the following theorem. T h e o r e m l . lt Let t be a real variable ranging over afnite or infnite interval (a,b) and z a complex variable ranging over a domain D. Assume that the function f (z, t ) satisfies the following conditions:

(i) f (z, t ) is a continuous function of both variables. (ii) For eachfxed value oft, f(z, t ) is a holomorphic function of z.