The question form the book is below.

I got as far as converting the f(z) eqn using Laurent's expansion to:

f(z) = 1/z - 3/(2z^3) + 11/(8z^5) + ...

Im stuck there, I took a look at the rest of the parts and haven't got the slightest clue how to do them, could anyone help me solve it, or point me in the right direction, as this book is full of similar qns which I want to be able to practise on after I learn how to do this.

Thanks in advance

It is required to evaluate the integral:

I = Integrate [1,-1] (1 - x^2)^1/2 / (1 + x^2) dx

in two different ways using contour integration.

(a) First consider the function f(z) = (z^2 - 1)^1/2 / (z^2 + 1)

[You are given that, with the choice -pi < arg(z+-1) <= pi, the only branch cut required is the section [-1,1] of the real axis]

(i) Find the value of

J = Integrate around a closed curve f(z) dz

around a circle of large radius

(ii) Relate J to the value of

K = Integrate around a closed curve f(z) dz

around a cut

(iii) Express K in terms of I, and hence evaluate I.