This is an interesting exercise though the wording is vague (perhaps deliberately in order to test your ability?). It asks to prove f(z) analytic at the origin if it is complex smooth. In other words, if the Cauchy formula (and hence the origin-shifted Cauchy formula) is valid for f(z), prove it has nth derivative. At the level of formal expressions, one could simply prove that the first derivative exists.

Derivative is defined by the limit: '
At the origin: '. Substitute Cauchy and origin-shifted cauchy formulas to the fraction, and get immediately the expression for the first derivative.

I agree that the wording in the book about this Exercise is vague. This is what I think Penrose wants you to do:

Taylor's expansion for about the point is:



Also Cauchy's formula in the origin shifted form is:



Show that is indeed given by:



I am about to substitute the Taylor series for f(z) given above into the contour integral without worrying about the rigorous justification, which is what Penrose suggests you do in the exercise:



and now a further step without rigorous justification



and then expanding the summation



Using the result of exercise 7.1 that when is an integer other than we can see that the above simplifies to

since see RTR book section 7.2

which was to be shown.

Nah. I don't think that's what the excercise meant. The stuffs you've put in are fine, but they are more or less covered the previous excercise. In fact, they're just the reverse of exercise 7.2.

Your starting point is wrong and you applied the wrong origin-shifted Cauchy because you haven't got the nth derivative established at this point yet. You need to prove its existence first; and the way forward is to derive the 1st derivative. All higher derivatives are then analogically derived.

As I said the text is vague and appears in reversed order.

As I undersand the excercise, we are supposed to show that the McLaurin series actually converges to f(z) when f^(n) (0) is defined by the Cauchy formula.

I am new on complex analysis, and do not know how to do that, but I think that you both misinterpreted what is being asked. I agree that the wording is pretty vague though.

A bit vague (and confused) around the exercise there. Er.. R. Penrose?
See your point. Like my History teacher used to say: exercising alot only turns one into an exercise slave rather than a master (who actually came up with the exercise). He's right, but useless. Unfortunately, unless one were one of the A's (Arnold/Albert) at birth, exercise's the only way to get there. Sad, eh?

This is really an exercise to our brain. I am trying to understand these equations. Anyone could help me? Hope I will get the point of it soon.