Complex Variables


Question 1
The value of k that makes the complex-valued function

f(z)=e^{-kx}(\cos 2y -i \sin 2y)

analytic, where z=x+iy, is _________. (Answer in integer)
A
1
B
2
C
3
D
4
GATE ME 2023   Engineering Mathematics
Question 1 Explanation: 
\begin{aligned} & f(z)=e^{-k x} \cos 2 y-i e^{-k x} \sin 2 y \\ & \text { Suppose }=e^{-k x} \cos 2 y=4(x, y) \\ & =-i e^{-k x} \sin 2 y=v(x, y) \end{aligned}
If function is analytical then it satisfy the equation
\begin{aligned} & \frac{\partial u}{\partial x}=-k e^{-k x} \cos 2 y &...(i)\\ & \frac{\partial u}{\partial y}=-2 e^{-k x} \sin 2 y &...(ii)\\ & \frac{\partial v}{\partial x}=-k e^{-k x} \sin 2 y &...(iii)\\ & \frac{\partial v}{\partial y}=-2 e^{-k x} \cos 2 y &...(iv) \end{aligned}
Cauchy-Riemann equation \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}
by putting equation (i), (iv) and solve then k=2
Question 2
Let w^{4}=16 j. Which of the following cannot be a value of w ?
A
2 e^{\frac{j 2 \pi}{8}}
B
2 e^{\frac{j \pi}{8}}
C
2 e^{\frac{j 5 \pi}{8}}
D
2 e^{\frac{j 9 \pi}{8}}
GATE EC 2023   Engineering Mathematics
Question 2 Explanation: 
w=(2) j^{1 / 4}
w=2(0+j)^{1 / 4}
w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4} =2\left[e^{j(2 n+1) \pi / 8}\right]

For n=0, w=e^{j \pi / 8}
For n=2, w=2 e^{5 \pi j / 8}
For n=4, w=2 e^{9 \pi j / 8}


Question 3
Given z=x+iy,i=\sqrt{-1} 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}\int_{c}^{}\frac{1}{(z-i)(z+4i)}dz is _______(round off to one decimal place).
A
0.2
B
0.4
C
0.6
D
0.8
GATE ME 2022 SET-2   Engineering Mathematics
Question 3 Explanation: 
Given contour is a circle at centre (0,0) and radius 2 given function is \frac{1}{(z-i)(z+4i)} here z = i is a singular point lies now by caucus integral formula

\begin{aligned} &\int \frac{1}{(z-i)(z+4i)}dz=\int \frac{\left ( \frac{1}{z+4} \right )}{z-i}dz=2 \pi i \times f(i)\\ &f(z)=\frac{1}{z+4i}\\ &f(i)=\frac{1}{5i}\\ &2 \pi i \times f(i)=2 \pi i \times \frac{1}{5i}=\frac{2\pi}{5}\\ &\text{Now }\frac{1}{2 \pi}\int_{c}\frac{1}{(z-1)(z+4i)}dz=\frac{1}{2 \pi}+\frac{2 \pi}{5}=\frac{1}{5} \end{aligned}
Question 4
Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?

A
\int \int _R dxdy
B
\oint _c xdy
C
\oint _c ydx
D
\frac{1}{2}\oint _c( xdy-ydx)
GATE EE 2022   Engineering Mathematics
Question 4 Explanation: 
Using green theorem?s
\oint _cF_1dx+F_2dy=\int \int _R\left ( \frac{\partial F_2}{\partial x} -\frac{\partial F_1}{\partial y}\right )dxdy
Check all the options:
\oint xdy=\int \int _R\left ( \frac{\partial x}{\partial x} -0\right )dxdy=\int \int _Rdxdy
\frac{1}{2}\oint xdy-ydx=\frac{1}{2}\int \int _R(1+1)dxdy=\int \int _Rdxdy
\oint ydx=\int \int _R(-1)dxdy=-\int \int _Rdxdy
Hence, \oint ydx is not represent the area of the region.
Question 5
Consider the following series:
\sum_{n=1}^{\infty }\frac{n^d}{c^n}
For which of the following combinations of c,d values does this series converge?
A
c=1,d=-1
B
c=2,d=1
C
c=0.5,d=-10
D
c=1,d=-2
GATE EC 2022   Engineering Mathematics
Question 5 Explanation: 
\begin{aligned} \Sigma u_n&=\Sigma \frac{n}{2^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{n+1}{2^{n+1}} \times \frac{2^n}{n}=\frac{1}{2}\\ \frac{1}{2}& \lt 2 \end{aligned}
\therefore By ratio test, \Sigma u_n is convergent.
(A) c=1, d=-1
\Sigma u_n=\Sigma \frac{1}{n} is divergent by P-test
(B) c=0.5, d=-10
\begin{aligned} \Sigma u_n&=\Sigma \frac{n^{-10}}{(0.5)^n}\\ &\text{Ratio test; }\\ \lim_{n \to \infty }\frac{u_{n+1}}{u_n}&=\lim_{n \to \infty }\frac{(n+1)^{-10}}{(0.5)^{n+1}} \times \frac{(0.5)^n}{n^{10}}\\ &=\frac{1}{0.5}=2\\ 2& \gt 1 \end{aligned}
\therefore By ratio test, \Sigma u_n is divergent.
(D) c=1,d=-2 \Sigma u_n=\Sigma \frac{n^{-2}}{(1)^n}=\Sigma \frac{1}{n^2}
\Sigma u_n is convergent by P-test.




There are 5 questions to complete.

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