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].  

Demonstration for Theorem 6.11.

[Graphics:../Images/IntegralRepresentationMod_gr_75.gif]

[Graphics:../Images/IntegralRepresentationMod_gr_76.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell