← 2019 Paper 2

UPSC 2019 Maths Optional Paper 2 Q1d — Step-by-Step Solution

10 marks · Section A

Cauchy-Riemann equations (necessary and sufficient) · Complex Analysis · asked 5× in 13 yrs · Read the full method →

Question

Suppose $f(z)$ is an analytic function on a domain $D$ in $\mathbb C$ and satisfies the equation $\operatorname{Im} f(z)=(\operatorname{Re} f(z))^2,\ z\in D$. Show that $f(z)$ is constant in $D$.

Technique

Cauchy–Riemann equations applied to the constraint $v=u^2$; the algebraic factor $1+4u^2>0$ forces all first partials to vanish on the connected domain.

Solution

Write $f=u+iv$ with $u=\operatorname{Re}f$, $v=\operatorname{Im}f$ real-valued and $C^\infty$ (analytic $\Rightarrow$ harmonic, hence smooth). The hypothesis is

Step 1 — Cauchy–Riemann equations

Analyticity gives

Step 2 — Differentiate the relation $v=u^2$

Substitute into (CR):

Step 3 — Solve the linear system for $(u_x,u_y)$

From the two equations,

Substitute the second into the first:

Since $1+4u^2\ge 1>0$ everywhere, we must have $u_x=0$. Then $u_y=-2u\,u_x=0$ as well.

Step 4 — Conclude

$u_x=u_y=0$ on the domain (connected open set) $D$, so $u$ is constant on $D$. By (CR), $v_x=-u_y=0$ and $v_y=u_x=0$, so $v$ is constant too (consistently, $v=u^2=$const). Hence

Answer