# Complex analysis: strategies for guessing harmonic conjugates

I have questions about two consecutive (and supposedly related) problems from my tutorial problem set:

1. Suppose $$u, v$$ are harmonic and satisfy the Cauchy-Riemann equations in $$\mathbb{R}^2$$. Show that $$f = u + iv$$ satisfies $$f'(x) = u_x(x, 0) − i u_y(x, 0) \tag{1}$$ for real $$x$$.

It seems to me that $$u$$ and $$v$$ being harmonic is irrelevant: if they satisfy the Cauchy-Riemann equations, are they not harmonic anyway? In any case, how are you supposed to show $$(1)$$ without using the definition of the derivative?

1. Using the result of the previous question to guess $$f$$, or otherwise, show that the following are harmonic and find analytic functions $$f$$ of which they are the real parts:
$$\quad$$ a) $$\ x - x^3 + 3xy^2$$
$$\quad$$ b) $$\ \cos(x) \cosh(y)$$
$$\quad \, \, \vdots$$

For part $$a)$$, my tutor jumped straight to $$z - z^3$$, and I see why he did; what I don't see is how $$(1)$$ is relevant. If we know that $$f'(x) = 1 - 3x^2 ,$$ so what?

In the first example we get $$f'(x)=1-3x^{2}$$ By basic Complex Analysis the fact that the holomorphic function $$f'(z)$$ coincides with $$c+z-z^{3}$$ for $$z$$ real (where $$c$$ is a constant) implies that the same holds for complex $$z$$ also. The harmonic conjugate of $$u$$ is therefore the imaginary part of $$c+z-z^{3}$$.
Similarly, in the second example, we get $$f'(x)=c+\cos x$$ so $$f'(z)=c+\cos z$$ and $$v(x,y)= \Im (c+\cos z) =a+\sin x \sinh y$$ where $$a =\Im c$$.
• (In my original post I asked if $u, v$ satisfying C-R $\implies u, v$ harmonic, not the other way around.) "...implies that the same holds for complex $z$ also." Why is this? Do I need to understand the proof for question 52 first? Commented Oct 26, 2020 at 10:19