Find conjugate harmonic function of $u=x^2+y^2$?
By Cauchy-Riemann equations,
$u_x=v_y$ and $u_y=-v_x$.
Now, $v_x=-u_y=-2y$ and $v_y=u_x=2x$.
We have $dv=\frac{\partial v}{\partial x} dx+\frac{\partial v}{\partial y} dy$.
$\implies dv=-2ydx+2xdy$. How can I find $v$?
Best Answer
The function $u$ is not harmonic, I'm pretty sure you wanted to write $y=x^2-y^2$. In that case by the Cauchy-Riemann equations:$$v_y=2x\implies v=2xy+\phi(x), $$ also we have that $$2y=2y+\phi'(x) \implies \phi(x)=C.$$ So the harmonic conjugate is $v(x,y) =2xy+C.$$