Composition of harmonic conjugate pairs
= Composition of harmonic conjugate pairs
If $u+iv$ and $p+iq$ are <holomorphic functions>, with the image of the first inside the domain of the second, then $p(u,v)+iq(u,v)$ is holomorphic by the <chain rule>. Directly, their <Cauchy-Riemann equations> imply $[p(u,v)]_x=[q(u,v)]_y$ and $[p(u,v)]_y=-[q(u,v)]_x$. Differentiating these relations proves harmonicity of both parts. No nonvanishing derivative hypothesis is required.