Chapter 12.4, Problem 83E

Laplace’s equation A classical equation of mathematics is Laplace’s equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensions, Laplace’s equation is

     $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0.$

Show that the following functions are harmonic; that is, they satisfy Laplace’s equation.

83.     $u\left(x,y\right)={\tan }^{-1}\left(\frac{y}{x-1}\right)-{\tan }^{-1}\left(\frac{y}{x+1}\right)$