SOS: 9367 Q65

[doctor723]

I really got stuck with this question...guess it is not difficult, but i haven't done enough complex analysis...could anyone help me out? I am gonna take sub in 21, April!!!

If f is a function defined by a complex power series expansion in z-a which converges for |z-a|<1 and diverges for |z-a|>1, which of the following must be true?
A. f(z) is analytic in the open unit disk with center at a
B. The power series for f(z+a) converges for |z+a|<1
C. f'(a)=0
D. Integral f(z)dz over c=0 for any circle C in the plane
E. f(z) has a pole of order 1 at z=a

Answer: A
My thought is to apply the ratio test for convergence, and got lim f(n+1)/f(n)=1, but how can I get the answer and exclude other choices? Plzzzzzzzz help!

[mstrfrdmx]

A satisfactory answer is dependent on your choice of definitions; using wikipedia's, the answer is trivial. http://en.wikipedia.org/wiki/Analytic_f ... efinitions

Some authors take $$f$$ analytic at $$x$$ to mean it is differentiable in an open neighborhood of $$x$$. It turns out the choice of definitions doesn't matter much. http://en.wikipedia.org/wiki/Analyticit ... _functions

[qwertyuiop]

I was badly struggling with Complex Analysis part but after reading through a textbook it became very clear to me.
You might have also got the similar confusion about analyticity and hope this helps:

If f is analytic at a point 'a', it means that f is analytic in an open disc containing 'a'. (so it's not necessarily differentiable at x=a in terms of the conventional differentiablity in the Real).

Also, here is a useful theorem:
Let f be analytic in an open disc D except for a finite number of exceptional points in D then for any closed curve, say gamma, in D that is not passing through any of the exceptional points the integral over gamma of f is 0.

[blitzer6266]

That isn't true. You need the region to be simply connected, or at least the curve has to be able to homotopy retract to a point.