Answer to Question #142497 in Complex Analysis for Doll

Question

Prove that u(x,y) given by the following is harmonic obtain it's corresponding conjugate and original function f(z)
u(x,y)=x^2-y^2

Answer 1

$\displaystyle u(x, y) = x^2 - y^2\\ \frac{\partial u}{\partial x} = 2x, \frac{\partial^2 u}{\partial x^2} = 2\\ \frac{\partial u}{\partial y} = -2y, \frac{\partial^2 u}{\partial y^2} = -2\\ \textsf{Since}\, \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \, u\, \textsf{is harmonic}\\ \textsf{Since a harmonic function is analytic}\\ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\\ 2x = \frac{\partial v}{\partial y}\\ v = \int 2x\, \mathrm{d}y\\ v = 2xy + C\\ \therefore \textsf{The complex conjugate is}\, 2xy + C\\ \therefore\, f(z) = u + jv = (x^2 - y^2) + j2xy + C= (x + jy)^2 + C = z^2 + C\\ \textsf{Where}\, C \, \textsf{is an arbitrary constant obtained}\\\textsf{from the integration process}$