Methods for finding harmonic conjugate function

Question

What are the methods for finding harmonic conjugate function? There is the Cauchy-Riemann equations but are there any other methods?

Answer 1

i am quoting from Ahlfors Complex Analysis, 2nd ed. p 27: Let u be the given function. Let v be the sought for function and let f=u+iv be the analytic function. Then f(z)=2u(z/2,z/(2i))-u(0,0)+constant. Then we find v easily

Answer 2

There is an elegant method proposed by A. Oppenheim. If $$ u\left ( \frac{1}{2}\cdot(z+z*) ,\frac{1}{2i}\cdot(z-z*) \right ) = \phi (z)+\psi (z*) $$ then $$ v(x,y)=-i\left \{ \phi (z)-\psi(z*)+c \right \} $$ adding these two together we get $$ f(z) = u + iv = 2\phi (z)+c $$

example, if: $$ u(x,y) = e^{x}cos(y) = \frac{1}{2}e^{x}(e^{iy}+e^{-iy})=\frac{1}{2}(e^{z}+e^{z*}) $$ then $$ f(z) =e^{z}+c $$

To derive this method one must show: $$ \frac{\partial u}{\partial x} = \frac{\partial \phi }{\partial z}+\frac{\partial \psi }{\partial z*} $$ $$ \frac{\partial u}{\partial y} = i\left ( \frac{\partial \phi }{\partial z}-\frac{\partial \psi }{\partial z*}\right ) $$ and then use the Cauchy-Riemann equations.