Complex Analysis [IAS (Admin.) Mains Mathematics]: Questions 1 - 7 of 39

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Question 1

Appeared in Year: 2016

Describe in Detail Subjective▾

Is is a harmonic function? Prove your claim. If yes, find its conjugate harmonic function and hence obtain the analytic function whose real and imaginary parts are and respectively. [CS (Main) Paper 2]

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Explanation

  • Given

… (131 more equations) …

Question 2

Appeared in Year: 2016

Describe in Detail Subjective▾

Prove that every power series represents an analytic function inside its circle of convergence. [CS (Main) Paper 2]

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Explanation

  • Suppose

  • Let be the radius of c…

… (495 more equations) …

Question 3

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 4

Appeared in Year: 2012

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Use Cauchy integral formula to evaluate , where is the circle . (CS main Paper 1)

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Explanation

Cauchy integral formula for Higher Derivative:- If a function is analytic within and on a closed contour and in any point within then derivate of all orders are analytic and are given by.

… (65 more equations) …

Question 5

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Solve the following linear programing problem by the simplex method. Write its dual. Also, write the optimal table of the dual from the optimal table of the given problem:

Subject to:

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Explanation

… (981 more equations) …

Question 6

Appeared in Year: 2011

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Let be a Non-singular, square Matrix. Show that . Hence show that . (CS main Paper 1)

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Explanation

Given be a Non-singular, square Matrix.

So, there exist of such that

… (175 more equations) …

Question 7

Appeared in Year: 2012

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Expand the function in Laurent series valid for

(CS main Paper 1)

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Explanation

Given

… (235 more equations) …