Theorem 6.11 (Leibniz's Rule).  Let G be an open set,  and let    be an interval of real numbers.  Let and its partial derivative with respect to z be continuous functions for all z in G and all t in I.  Then  

               is analytic for z in G, and  

            .  

Proof.

    The proof is presented in some advanced texts.  See, for instance, Rolf Nevanlinna and V. Paatero, Introduction to Complex Analysis (Reading, Massachusetts: Addison-Wesley Publishing Company, 1969), Section 9.7.

Complex Analysis for Mathematics and Engineering