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

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## Question number: 27

Appeared in Year: 2006

### Write in Short

Evaluate

## Question number: 28

» Differentiability in Complex Plane

Appeared in Year: 2011

### 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

Appeared in Year: 2017

### 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

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## Question number: 30

» Taylor Series Expansion (Complex)

Appeared in Year: 2015

### Describe in Detail

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

### Explanation

So that

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## Question number: 31

Appeared in Year: 2005

### Describe in Detail

What is the Radius of Convergence of the power series

?

### Explanation

The given power series is

Now

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## Question number: 32

» Leibniz Rule for Differentiation under Integral Sign

Appeared in Year: 2006

### 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

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## Question number: 33

Appeared in Year: 2010

### 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

Appeared in Year: 2008

### Write in Short

Evaluate: