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]
EditExplanation
- 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)
EditExplanation
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)
EditExplanation
Rouche՚s Theorem:- If and are analytic inside and on a simple closed curve and if on , then
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Question 4
Appeared in Year: 2015
Describe in Detail Subjective▾
State Cauchy՚s residue theorem. Using it, evaluate the integral
(Paper I)
EditExplanation
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)
EditExplanation
- Now
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Question 6
Appeared in Year: 2007
Describe in Detail Subjective▾
Evaluate (by using residue theorem)
(Paper 2)
EditExplanation
- Let
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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
Explanation
Consider the integral
Where
Taken round the closed…
… (389 more equations) …