1
GATE ME 2026
MCQ (Single Correct Answer)
+2
-0

If $w=\log _e z=\log _e(x+i y)$, where $i=\sqrt{-1}$, then which one of the following statements is correct?

A

$w$ is analytic everywhere except at $z=0$

B

$w$ is non-analytic everywhere

C

The conjugate functions of $w$ are $\log _e\left(x^2+y^2\right)$ and $\log _e\left(x^2-y^2\right)$

D

The conjugate functions of $w$ are $\tan ^{-1}(x / y)$ and $\tan ^{-1}(y / x)$

2
GATE ME 2025
Numerical
+2
-0

If $C$ is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is _______ (rounded off to 1 decimal place).

$$ \oint_c \frac{z^3}{\left(z^2+4\right)\left(z^2-4\right)} d z=2 \pi i n $$

Your input ____
3
GATE ME 2022 Set 2
Numerical
+2
-0
Given z = x +iy, i = √-1 C is a circle of radius 2 with the centre at the origin. If the contour C is traversed anticlockwise, then the value of the integral $\frac{1}{2\pi}\int_c\frac{1}{(z-i)(z+4i)}dZ$ is ________ (round off to one decimal place.)
Your input ____
4
GATE ME 2022 Set 1
MCQ (Single Correct Answer)
+2
-0.66

The value of the integral

$\rm \oint \left( \frac{6z}{2z^4 - 3z^3 + 7 z^2 - 3z + 5} \right) dz$

evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z = i, where 𝑖 is the imaginary unit, is

A
(-1 + i) π
B
(1 + i) π
C
2(1 - i) π
D
(2 + i) π

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