NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 50 - 56 of 101

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Question number: 50

» Residues of Complex Function

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Appeared in Year: 2009

Essay Question▾

Describe in Detail

Let C be the contour consisting of the lines , described counterclockwise in the plane. Compute

Explanation

is the contour shown in the figure

The contour consisting of the lines

The Contour Consisting of the Lines

The contour consisting of the lines

To evaluate

Now is a pole of order 1 (simple pole)

Also

… (45 more words) …

Question number: 51

» Taylor Series Expansion (Complex)

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Appeared in Year: 2017

Essay Question▾

Describe in Detail

Let Write down the Taylor Expansion of f in the neighbourhood of .

Explanation

Here

Put

… (94 more words) …

Question number: 52

» Tests of Convergence

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Appeared in Year: 2015

Essay Question▾

Describe in Detail

In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent

a)

b)

c)

Explanation

Option (a) The given series is

Where

Now comparing it with

1)

2

… (170 more words) …

Question number: 53

» Absolutely Convergent

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Appeared in Year: 2006

MCQ▾

Question

Pick out series which are absolutely convergent.

Choices

Choice (4) Response

a.

b.

c.

d.

Both a. and b. are correct

Question number: 54

» Sequences and Series (Convergence)

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Appeared in Year: 2005

Essay Question▾

Describe in Detail

What are the value of for which the following series is convergent?

Explanation

Let the omen series be denoted by ……. . eq. (1)

Where

Since for absolute convergence, we take

Now is convergent if …………eq. (2)

… (94 more words) …

Question number: 55

» Order of Pole and Its Residue

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Appeared in Year: 2010

Essay Question▾

Describe in Detail

Find the order of the pole and its residue at of the function

Explanation

Order of pole

… (119 more words) …

Question number: 56

» Line Integral in Complex Plane

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Appeared in Year: 2017

Essay Question▾

Describe in Detail

Let C be the contour in the complex plane consisting of two straight line segments, one from and the other from . Evaluate

Where

Explanation

Graph of the two straight line segments

Graph of the Two Straight Line Segments

Graph of the two straight line segments

Along OM

The imaginary axis from is the line segment

Now on so that

Also y varies from

… (103 more words) …