Let D be a region in C and let u(x,y)be a real-valued function on D. We seek another real-valuedfunction v(x,y) such that f(z) = u + iv is analytic, i.e. satisfiesthe Cauchy-Riemann equations.
Let D b e a region in C and let u(x, y) be a real-valued function on D. We seek another real-valued function v (x, y) such that f (z) = u + iv is analytic, i.e. satisfies the Cauchy-Riemann equations. Equivalently, we want to find a function v such that and , which says that v is the vector field F = ( ). Show that F satisfies the mixed partials condition exactly when u is harmonic. Conclude that, if D is simply connected, then F is a gradient vector field v and hence that u is the real part of an analytic function.
