Complex Analysis analytic function

Question

If $f$ is an analytic function on a domain $D$ and $\mathrm{Im} f$ takes on only the value $71$ then for some constant $C \in \Bbb{R}$, is it true that $f = C + 71 i$ on $D$?

Answer 1

The answer is yes.

Setting $f = u + iv$, look at the Cauchy-Riemann equations for $f$:

$u_x = v_y, \tag{1}$

$u_y = -v_x. \tag{2}$

Since $\text{Im} f = v = 71$, $v_x = v_y = 0$; thus by (1), (2) $u_x = u_y = 0$ as well. But this just says $\nabla u = 0$, so $u$ must be a real constant, call it $C \in \Bbb R$. Thus $f = C + 71i$.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!