Harmonic Conjugate Function

The harmonic conjugate to a given function $u(x,y)$ is a function $v(x,y)$ such that

$$f(x,y) = u(x,y)+v(x,y)$$

is Complex Differentiable (i.e., satisfies the Cauchy-Riemann Equations). It is given by

$$v(z) = \int u_x\,dy-u_y\,dx.$$