# How do I find the harmonic conjugate of a composite of two functions

Given an entire function $$f(z)$$ how do I find the harmonic conjugate of $$log(|f'(z)|)$$.

I tried directly solving the differential equations obtained via Cauchy–Riemann equations as follows:

set $$f(z) = u(x, y) + iv(x, y)$$ and let $$\psi(x, y) = log(|f'(z)|) = \frac{1}{2}log(u_x^2 + v_x^2) = \frac{1}{2}log(u_y^2 + v_y^2)$$ and hence, $$\psi_x = \frac{u_yu_{xy} + v_yv_{xy}}{u_y^2 + v_y^2}$$ and $$\psi_y = \frac{u_xu_{xy} + v_xv_{xy}}{u_x^2 + v_x^2}$$

I tried to further use that $$\psi_x = \phi_y$$ and $$\phi_x = -\psi_y$$ to solve for $$\phi(x, y)$$ however that was useless. How else should I proceed with finding the harmonic conjugate?

$$\log g=\log |g|+i\arg g$$ whenever RHS makes sense. (since $$arg$$ is multivalued, RHS may make sense only locally, so only near any point where $$g \ne 0$$, and it may be stitched together under certain circumstances like simple connectedness of the domain $$U$$ where $$g$$ doesn't vanish, $$\Re g >0$$ or even better but harder to check, $$g+ c \ne 0, c \ge 0$$ and the like)
So the conjugate harmonic of $$\log|f'|$$ is $$\arg f'$$ with the comments above that it may make sense only locally but not globally