GATE 2024 IN Q.49: solution and answer

GATE 2024 · INQ.49 · 2 marks · MSQ

The complex functions $f (z) = u (x, y) + i v (x, y)$ and $\overline{f (z)} = u (x, y) - i v (x, y)$ are both analytic in a given domain. Choose the correct option(s) from the following.

(A)
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} = 0$
(B)
$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \neq = 0$
(C)
$\frac{df ( z )}{d z} = 0$
(D)
$\frac{df ( z )}{d z} \neq = 0$

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Answer and solution

ANSWER

Options (A) and (C) are correct. Analyticity of both $f$ and $\overline{f}$ forces $u_{x} = v_{y} = 0$ and $u_{y} = v_{x} = 0$ , so $f^{'} (z) = u_{x} + i v_{x} = 0$ .

Official answer: A, CSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

Since $f = u + i v$ is analytic, it satisfies the Cauchy-Riemann equations: $u_{x} = v_{y}$ and $u_{y} = - v_{x}$ .
Write $\overline{f (z)} = u + i (- v)$ : its real part is $u$ and its imaginary part is $- v$ . Since $\overline{f}$ is analytic too, its Cauchy-Riemann equations give $u_{x} = (- v)_{y} = - v_{y}$ and $u_{y} = - (- v)_{x} = v_{x}$ .
Combine the first equations: $u_{x} = v_{y}$ and $u_{x} = - v_{y}$ give $2 v_{y} = 0$ , so $u_{x} = v_{y} = 0.$ This is option (A), so (A) is true.
Combine the second equations: $u_{y} = - v_{x}$ and $u_{y} = v_{x}$ give $2 v_{x} = 0$ , so $u_{y} = - v_{x} = 0.$ Both sides are zero. Option (B) says they are $\neq = 0$ , so (B) is false.
For an analytic function, $f^{'} (z) = u_{x} + i v_{x}$ . Here $f^{'} (z) = 0 + i \cdot 0 = 0$ at every point of the domain, so (C) is true and (D) is false. (All four first partial derivatives of $u$ and $v$ vanish, so on a connected domain $f$ is a constant.)
Correct options: (A) and (C).

KEY IDEA

Write the Cauchy-Riemann equations for $f = u + i v$ and for $\overline{f} = u + i (- v)$ . Each pair of equations has the same left side and right sides of opposite sign, so every first partial derivative of $u$ and $v$ must be zero. That makes $f^{'} (z) = u_{x} + i v_{x} = 0$ , and on a connected domain $f$ is constant.

WATCH OUT FOR

Using $+ v$ instead of $- v$ as the imaginary part of $\overline{f}$ in its Cauchy-Riemann equations. Doing that just repeats the equations for $f$ and gives no new information. Another trap is picking (B) because it looks like the standard CR equation $u_{y} = - v_{x}$ . The equation does hold, but both sides equal zero, so the " $\neq = 0$ " part of (B) is false.

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GATE 2024 IN Q.49: solution and answer

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