Theorem 3.9 (Construction of a Conjugate).  Let    be harmonic in an -neighborhood of the point .  Then there exists a conjugate harmonic function    defined in this neighborhood such that   is an analytic function.

Proof.  A conjugate harmonic function v will satisfy the Cauchy-Riemann equations and .  Assuming that such a function exists, we determine what it would have to look like by using a two-step process.  First, we integrate (which should equal ) with respect to y and get

(3-27)            

where is a function of x alone that is yet to be determined.  Second, we compute by differentiating both sides of this equation with respect to x and replacing    with    on the left side, which gives  

            

It can be shown (we omit the details) that because u is harmonic, all terms except those involving x in the last equation will cancel, revealing a formula for involving x alone.  Elementary integration of the single-variable function can then be used to discover .  We finally observe that the function v so created indeed has the properties we seek.

    The functions   and are computed with the formulas:

            ,   and     
            
            .  

Proof.

Proof of Theorem 3.9 is in the book.

Complex Analysis for Mathematics and Engineering