Question:medium

Which of the following function satisfies Cauchy-Riemann equations:

Show Hint

Note that $e^x \cdot e^{iy} = e^{x+iy} = e^z$, which is the standard complex exponential function $f(z) = e^z$. Any function depending purely on $z$ (and not $\bar{z}$) satisfies C-R equations!

Updated On: Jul 29, 2026

$x - iy$
$e^x \cdot e^{-iy}$
$e^y \cdot e^{ix}$
$e^x \cdot e^{iy}$

Show Solution

The Correct Option is D

Solution and Explanation

Was this answer helpful?

Top Questions on Complex Analysis

Match List-I with List-II and choose the correct option:

LIST-I (Function)	LIST-II (Value)
(A) $ \int_{\gamma} \frac{1}{z-a} \, dz $, where $ \gamma: \|z-a\|=r, r > 0 $	(III) $ 2i\pi $
(B) $ \int_{\gamma} \frac{z+2}{z} \, dz $, where $ \gamma: z = 2e^{it}, 0 \le t \le \pi $	(IV) $ i\pi $
(C) $ \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz $, where $ \gamma: \|z\|=3 $	(II) $ 2i\pi(e^4 - e^2) $
(D) $ \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz $, where $ \gamma: \|z\|=2 $	(I) $ -4 + 2i\pi $

Choose the correct answer from the options given below:

CUET (PG) - 2025
Mathematics
Complex Analysis

View Solution

Questions Asked in CUET (PG) exam

A trade-off is:

CUET (PG) - 2024
Trade-offs

View Solution

If a Production Function is given as Q(x,y) = Axay, where (a + b) = 2, the function is said to display:

CUET (PG) - 2024
Production Theory

View Solution

In an oligopolistic market, firm i and firm j have constant marginal cost = c for an identical good. They compete to set prices Pi and Pj. The demand for total market demand Q, where if $P_i > P_j$, the demand for firm i is 0. If $P_i < P_j$, the demand for firm i is Q. If $P_₁ = P_j$, then they share the market equally and hence the demand for firm i is $\frac{Q}{2}$. In equilibrium, the prices of firms i and j are:

CUET (PG) - 2024
Oligopoly

View Solution

If Y = F(L,K) is a linear homogeneous production function, then multiplying all inputs by a constant factor α = 2 results in output increasing by a factor of:

CUET (PG) - 2024
Production Function

View Solution

In the table below, firm 1 and firm 2 can cooperate to share market profits or compete with each other. The figures in the cells represent the profits in crores of INR. They can either cooperate with each other where they share INR 10 crores equally. Alternatively, they can compete, which lowers the market output to INR 4 crores, which they share equally. If one of the firms competes while the other cooperates, the former obtains INR 10 crore of profit while the other makes zero profit. In equilibrium, firm A and firm B should respectively:

Firm 2	Cooperate	Compete
Firm 1	5, 5	0, 10
Compete	10,0	2, 2

CUET (PG) - 2024
Game theory

View Solution

LIST-I (Function)	LIST-II (Value)
(A) \( \int_{\gamma} \frac{1}{z-a} \, dz \), where \( \gamma: \|z-a\|=r, r > 0 \)	(III) \( 2i\pi \)
(B) \( \int_{\gamma} \frac{z+2}{z} \, dz \), where \( \gamma: z = 2e^{it}, 0 \le t \le \pi \)	(IV) \( i\pi \)
(C) \( \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz \), where \( \gamma: \|z\|=3 \)	(II) \( 2i\pi(e^4 - e^2) \)
(D) \( \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz \), where \( \gamma: \|z\|=2 \)	(I) \( -4 + 2i\pi \)