Hi everybody,

Fisrt of all, I would like to apologize for my sometimes poor english, but english is not my native language.

Here is a solution of exercise 7.5.

Regards,

Your argument about the second integral applies to any analytic function of course, not just , so all you need to be concerned about is what happens to arg(z) going around the little curve.
Now try it with the big semicircle in the lower half plane :)
Laura

This is a brilliant proof by Fred. The key here is Cauchy integral theorem, where the integral on a closed contour is zero. The bloody book didn't say anything about this theorem at all.

By substituting z with (r,) the second and fourth integrals can be derived more quickly, without the need for integration by parts.

"This theorem" follows from Cauchy's formula which IS given in the book.

Well, Fred's solution starts by making use of the Residue theorem, which is given by exercise [7.4], and it also uses Cauchy's integral formula, which is given in section 7.3 of the book as vasco says.

Penrose didn't show that the integral around a contour is the sum of the integrals around contour segments, but he can only go so far, I guess.