On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's pretty straightforward. But what does she mean about this region and the derivatives. I get that there are higher order derivatives. But I wish she would have given an example of one such function. Also, isn't analytic here only tested for by taking the first derivative? So if we do that we can't just assume that there will be higher order derivatives up to the nth order.