Theorem 6.11 (Leibniz's Rule).  Let G be an open set,  and let  [Graphics:Images/IntegralRepresentationMod_gr_68.gif]  be an interval of real numbers.  Let [Graphics:Images/IntegralRepresentationMod_gr_69.gif] and its partial derivative [Graphics:Images/IntegralRepresentationMod_gr_70.gif] with respect to z be continuous functions for all z in G and all t in I.  Then  

            [Graphics:Images/IntegralRepresentationMod_gr_71.gif]   is analytic for z in G, and  

            [Graphics:Images/IntegralRepresentationMod_gr_72.gif].  

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