Complex analysis: strategies for guessing harmonic conjugates

Question

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

52. 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?

53. 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?

Answer 1

It is true that any two functions satisfying the C - R equations are harmonic (assuming that they are smooth).

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 $a =\Im c$.