English
Hindi

Question

Download Solution PDF

Let γ be the positively oriented circle in the complex plane given by {z ∈  ∶ |z - 1| = 1/2}. The line integral

 equals

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
Download PDF Attempt in App
View all CSIR NET Papers >
  1. iπe
  2. -iπe
  3. πe
  4. -πe

Answer (Detailed Solution Below)

Option 1 : iπe
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

Concept:

Cauchy Integral Theorem:

If a complex function f(z) is analytic within and on a closed contour C inside a simply-connected domain, and if a is any point in the middle of C, then

f(a) = 

Explanation:

Singular points are given by 

z2 - 1 = 0 ⇒ z = 1, z = -1

The only singular point that lies inside γ is z = 1.

Let f(z) = 

Hence using Cauchy's Integral test

 = 2πi f(1) = 2πi × (e/2) = iπe

 Option (1) is correct. 

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jul 19, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website. The exam will be held on 28th July 2025.

-> CSIR NET City Intimation Slip 2025 has been released @csirnet.nta.ac.in.

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

‹‹ Previous Ques
Next Ques ››
Exams Tests Super Practice

More Contour Integral & Theorem Questions

Q1.Let C be the positively oriented circle in the complex plane of radius 3 centered at the origin. What is the value of the integral \(\rm\displaystyle\int_C \frac{d z}{z^2\left(e^z−e^{−z}\right)}\)?
Q2.Let p be a positive integer. Consider the closed curve r(t) = eit, 0 ≤ t < 2π. Let f be a function holomorphic in {z ∶ |z| < R} where R > 1. If f has a zero only at z0, 0 < |z0| < R, and it is of multiplicity q, then  \(\frac{1}{2 π i} \int_r \frac{f^{\prime}(z)}{f(z)} z^p d z\)  equals 
Q3.Let γ be the positively oriented circle in the complex plane given by {z ∈ \(\mathbb{C}\) ∶ |z - 1| = 1/2}. The line integral \(\int_\gamma \frac{z e^{1 / z}}{z^2-1} d z\) equals

More Complex Analysis Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q3.Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define γr(t) = reit (where t ∈ |0, 2π|) and Ir = \(\rm \frac{1}{2\pi i}\int_{\gamma_r}\frac{z^2+1}{(z-a)(z-b)}dz\) Which of the following statements is necessarily true? 
Q4.Let f be an entire function. Which of the following statement is FALSE?
Q5.For a complex number a such that 0 < |a| < 1, which of the following statements is true?  
Q6.Let 𝔻 = { 𝑧 ∈ ℂ ∶ |𝑧| < 1} and 𝑓: 𝔻 → ℂ be defined by  \(f(z)=\rm z-25z^3+\frac{z^5}{5!}-\frac{z^7}{7!}+\frac{z^9}{9!}-\frac{z^{11}}{11!}\) Consider the following statements: 𝑃: 𝑓 has three zeros (counting multiplicity) in 𝔻. 𝑄: 𝑓 has one zero in 𝕌 = { 𝑧 ∈ ℂ ∶ \(\frac{1}{2}\) < |𝑧| < 1}. Then 
Q7.Let f be an entire function that satisfies |f(z)| ≤ ey for all z = x + iy ∈ ℂ, where x, y ∈ ℝ. Which of the following statements is true?
Q8.Let C be the positively oriented circle in the complex plane of radius 3 centered at the origin. What is the value of the integral \(\rm\displaystyle\int_C \frac{d z}{z^2\left(e^z−e^{−z}\right)}\)?
Q9.Consider the function f defined by f(z) = \(\rm\frac{1}{1−z−z^2}\) for z ∈ ℂ such that 1 − z − z2 ≠ 0. Which of the following statements is true?

More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti boss teen patti cash game yono teen patti