NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 27 - 36 of 101

(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…

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

So that

This is the Required power series Expansion of the function in the neighbourhood of .

The given power series is

Now

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.

Here let

Here is a simple pole and lies within the circle

By Cauchy residue theorem

In triangle OAB

area of triangle

Area of polygon whose vertices are represented in the complex plane by the eighth roots of unity