UPSC 2024 Maths Optional Paper 2 Q1b — Step-by-Step Solution

10 marks · Section A

Harmonic functions and harmonic conjugate · Complex Analysis · asked 7× in 13 yrs · Read the full method →

Question

If $w=f(z)$ is an analytic function of $z$ , show that

\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\log|f'(z)|=0.

Identify $\log|f'(z)|$ as the real part of the analytic function $\log f'(z)$ ; real parts of analytic functions are harmonic.

Step 1 — Construct an analytic function.

Suppose $f'(z)\ne 0$ . Locally, $\log f'(z)$ is an analytic function (composition of the analytic derivative $f'$ with a local branch of $\log$ ):

\log f'(z)=\log|f'(z)|+i\arg f'(z).

The real part of this analytic function is $U(x,y)=\log|f'(z)|$ .

Step 2 — Real parts of analytic functions are harmonic.

If $G(z)=U+iV$ is analytic, then $U$ satisfies the Cauchy–Riemann equations and consequently $U_{xx}+U_{yy}=0$ .

Applying this to $G(z)=\log f'(z)$ :

\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\log|f'(z)|=0.

\boxed{\;\log|f'(z)|\text{ is harmonic wherever }f'\ne 0.\;}