Complex Analysis-Cauchy's Residue Theorem [IAS (Admin.) Mains Mathematics]: Questions 1 - 8 of 8

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Question 1

Appeared in Year: 2016

Describe in Detail Subjective▾

Let be the curve find, giving justifications, the value of the contour integral

[CS (Main) Paper 2]

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Explanation

  • Given represents a circle with radius 1.

    Now

    Let

… (82 more equations) …

Question 2

Appeared in Year: 2015

Describe in Detail Subjective▾

State Cauchy՚s Residue Theorem using it Evaluate the integral.

(CS main Paper 2)

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Explanation

Cauchy Residue Theorem:- If f is analytic in a domain D expect for isolated singularities at , then, for any closed center r in D on which none of the point lie, we have

… (646 more equations) …

Question 3

Appeared in Year: 2013

Describe in Detail Subjective▾

Prove that if where positive and real, then the function are has Zeros in the unit circle. (CS main Paper 1)

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Explanation

Rouche՚s Theorem:- If and are analytic inside and on a simple closed curve and if on , then

… (239 more equations) …

Question 4

Appeared in Year: 2015

Describe in Detail Subjective▾

State Cauchy՚s residue theorem. Using it, evaluate the integral

(Paper I)

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Explanation

Let

Poles of are given by

… (690 more equations) …

Question 5

Appeared in Year: 2013

Describe in Detail Subjective▾

Using Cauchy՚s residue theorem, evaluate the integral

(Paper II)

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Explanation

  • Now

… (151 more equations) …

Question 6

Appeared in Year: 2007

Describe in Detail Subjective▾

Evaluate (by using residue theorem)

(Paper 2)

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Explanation

  • Let

… (154 more equations) …

Question 7

Appeared in Year: 2021

Describe in Detail Subjective▾

Using contour integration, evaluate to integral

Edit

Explanation

Consider the integral

taken round the contour C consisting of

i. The real axis from P to R,

… (190 more equations) …

Question 8

Appeared in Year: 2022

Describe in Detail Subjective▾

Apply the calculus of residues to evaluate

Edit

Explanation

Consider the integral

Where

Taken round the closed…

… (389 more equations) …