NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 27 - 36 of 101
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Question 27
Question 28
Appeared in Year: 2011
Describe in Detail
Essay▾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
The points z in the complex plane where exist are those for which Cauchy – Riemann conditions are satisfied
i.e.
i.e.
Thus the points where exist are
(b) Let
Now all these partial derivatives exist and continuous functions of
The points where exists are those wher…
… (4 more words) …
Question 29
Appeared in Year: 2017
Describe in Detail
Essay▾Let , be a fixed positive integer. Let for
Evaluate:
Explanation
Numerical value of last term of
i.e.
And
Here
The given series is
Differentiating both sides with respect to , we get
Putting , then we get
Thus the answer is zero
Alternative Method
L. H. S
in bracket put , then
Question 30
Appeared in Year: 2015
Describe in Detail
Essay▾Write down the power series expansion of the function in a neighbourhood of .
Explanation
So that
This is the Required power series Expansion of the function in the neighbourhood of .
Question 31
Appeared in Year: 2005
Describe in Detail
Essay▾What is the Radius of Convergence of the power series
?
Explanation
The given power series is
Now
Question 32
Appeared in Year: 2006
Describe in Detail
Essay▾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
[Using Leibnitz Rule]
Which is a function of x.
Question 33
Appeared in Year: 2010
Describe in Detail
Essay▾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
By Cauchy residue theorem
Question 34
Question 35
Appeared in Year: 2012
Question
MCQ▾Let be a sequence of functions defined on . Determine . For each the following.
Choices
Choice (4) | Response | |
---|---|---|
a. | ||
b. | ||
c. | ||
d. | All a., b. and c. are correct |
Question 36
Appeared in Year: 2013
Describe in Detail
Essay▾Find the area of the polygon whose vertices are represented in the complex plane by the eighth roots of unity.
Explanation
In triangle OAB
area of triangle
Area of polygon whose vertices are represented in the complex plane by the eighth roots of unity