UPSC 2024 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

If $\phi$ and $\psi$ are functions of $x$ and $y$ satisfying Laplace’s equation, show that $f(z)=p+iq$ is an analytic function, where $p=\dfrac{\partial\phi}{\partial y}-\dfrac{\partial\psi}{\partial x}$ and $q=\dfrac{\partial\phi}{\partial x}+\dfrac{\partial\psi}{\partial y}$ .

Technique

Directly verify the Cauchy–Riemann equations $p_x=q_y$ and $p_y=-q_x$ ; the harmonic conditions on $\phi,\psi$ provide the cancellations.

Solution

Compute partial derivatives with mixed partials treated as equal (Clairaut’s theorem, valid under the smoothness implied by harmonicity):

p_x=\phi_{yx}-\psi_{xx},\quad p_y=\phi_{yy}-\psi_{xy},\quad q_x=\phi_{xx}+\psi_{yx},\quad q_y=\phi_{xy}+\psi_{yy}.

Check $p_x=q_y$ :

p_x-q_y=(\phi_{xy}-\psi_{xx})-(\phi_{xy}+\psi_{yy})=-(\psi_{xx}+\psi_{yy})=0,

since $\psi$ is harmonic.

Check $p_y=-q_x$ :

p_y+q_x=(\phi_{yy}-\psi_{xy})+(\phi_{xx}+\psi_{xy})=\phi_{xx}+\phi_{yy}=0,

since $\phi$ is harmonic.

Both Cauchy–Riemann equations are satisfied, so $f(z)=p+iq$ is analytic.

Answer

\boxed{\;f(z)=p+iq\text{ is analytic wherever }\phi\text{ and }\psi\text{ are harmonic.}\;}