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γ को {z ∈ \(\mathbb{C}\) ∶ |z - 1| = 1} से दिए जाने वाले सम्मिश्र समतल में धनात्मक रूप से अभिविन्यस्त वृत्त मान लें। तब \(\frac{1}{2 \pi i} \int_γ \frac{d z}{z^3-1}\) निम्न के बराबर है

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Last updated on Jun 23, 2025

-> The last date for CSIR NET Application Form 2025 submission has been extended to 26th June 2025.

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

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-> 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.

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More Contour Integral & Theorem Questions

Q1.मानें कि सम्मिश्र समतल में, C एक धनात्मक रूप से अभिविन्यस्त वृत्त है जो मूल बिंदु पर केंद्रित एवं त्रिज्या 3 का है। तब समाकलन \(\rm\displaystyle\int_C \frac{d z}{z^2\left(e^z−e^{−z}\right)}\) का क्या मान है?  
Q2.मानें कि p एक धनात्मक पूर्णांक है। संवृत वक्र r(t) = eit, 0 ≤ t < 2π पर विचार करें। मानें कि f ऐसा फलन है जो {z ∶ |z| < R} में सममितीय (होलामॉर्फिक) है जहां R > 1 है। यदि f के शून्य केवल z0 में हो, z0 ≠ 0, |z0| < R, और उसकी बहुकता (multiplicity) q हो, तब \(\frac{1}{2 π i} \int_r \frac{f^{\prime}(z)}{f(z)} z^p d z\) का मान निम्न है
Q3.γ को {z ∈ \(\mathbb{C}\) ∶ |z - 1| = 1/2} से दिये जाने वाले सम्मिश्र समतल में धनात्मक रूप से अभिविन्यस्त वृत्त मान लें। तब रैखीय समाकलन \(\int_\gamma \frac{z e^{1 / z}}{z^2-1} d z\) निम्न के बराबर है

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 f be an entire function. Which of the following statement is FALSE?
Q4.For a complex number a such that 0 < |a| < 1, which of the following statements is true?  
Q5.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 
Q6.मानें कि f ऐसा सर्वत्र वैश्लेषिक फलन है जो |f(z)| ≤ ey को सभी z = x + iy ∈ ℂ, जहां x, y ∈ ℝ, के लिए संतुष्ट करता है। निम्न वक्तव्यों में से कौन-सा बक्तव्य सत्य है?
Q7.मान लीजिए f(z) = exp\(\rm\left(z+\frac{1}{z}\right)\), z ∈ ℂ\{0} है। तब f का z = 0 पर अवशेष _______ है।
Q8.मानें कि सम्मिश्र समतल में, C एक धनात्मक रूप से अभिविन्यस्त वृत्त है जो मूल बिंदु पर केंद्रित एवं त्रिज्या 3 का है। तब समाकलन \(\rm\displaystyle\int_C \frac{d z}{z^2\left(e^z−e^{−z}\right)}\) का क्या मान है?  
Q9.ऐसे z ∈ ℂ कि 1 − z − z2 ≠ 0 हो, के लिए f(z) = \(\rm\frac{1}{1−z−z^2}\) द्वारा परिभाषित फलन f पर विचार करें। निम्न वक्तव्यों में से कौन सा सत्य है?

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? 
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