Theorem 3.8.  Let    be an analytic function on a domain D.  Then both    and    are harmonic functions on D.  In other words, the real and imaginary parts of an analytic function are harmonic.

Proof.  In Corollary 6.3 we will show that, if f(z) is analytic, then all partial derivatives of u and v are continuous.  Using that result here, we see that, as f is analytic, u and v satisfy the Cauchy-Riemann equations

           and   .  

Taking the partial derivative with respect to x of each side of these equations gives  

           and   .  


Similarly, taking the partial derivative of each side with respect to y yields  

           and   .  


The partial derivatives are all continuous, so we use a theorem from the calculus of real functions that states that the mixed partial derivatives are equal; that is,  

           and   .  


Combining all these results finally gives

        ,  and  

        .  

Therefore both u and v are harmonic functions on D.

Proof.

Proof of Theorem 3.8 is also in the book.

Complex Analysis for Mathematics and Engineering