NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 27 - 34 of 101

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Question number: 27

» Cauchy Residue Theorem

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Appeared in Year: 2006

Short Answer Question▾

Write in Short

Evaluate

Question number: 28

» Differentiability in Complex Plane

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Appeared in Year: 2011

Essay Question▾

Describe in Detail

Find the points z in the complex plane where exists and evaluate it at those points

a)

b) , Where denotes the imaginary part of z.

Explanation

(a) Let

Clearly all the four partial derivatives exist and continuous functions of

… (149 more words) …

Question number: 29

» Binomial Series

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Appeared in Year: 2017

Essay Question▾

Describe in Detail

Let , be a fixed positive integer. Let for

Evaluate:

Explanation

Numerical value of last term of

i. e

And

Here

The given series is

… (114 more words) …

Question number: 30

» Taylor Series Expansion (Complex)

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Appeared in Year: 2015

Essay Question▾

Describe in Detail

Write down the power series expansion of the function in a neighbourhood of .

Explanation

So that

… (109 more words) …

Question number: 31

» Radius of Convergence

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Appeared in Year: 2005

Essay Question▾

Describe in Detail

What is the Radius of Convergence of the power series

?

Explanation

The given power series is

Now

… (93 more words) …

Question number: 32

» Leibniz Rule for Differentiation under Integral Sign

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Appeared in Year: 2006

Essay Question▾

Describe in Detail

Let . Define Evaluate as a function of x.

Explanation

Leibnitz Rule is given by

If the function are defined on and are differentiation at a point is continuous, then

… (142 more words) …

Question number: 33

» Cauchy Residue Theorem

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Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let C denote the circle in the complex plane, described in the positive i. e (anti-clockwise) sense. Evaluate

Explanation

Here let

Here is a simple pole and lies within the circle

… (93 more words) …

Question number: 34

» Trigonometric Identities

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Appeared in Year: 2008

Short Answer Question▾

Write in Short

Evaluate: