Rigorous proof of higher derivatives form Cauchy Integral formula.

Rigorous proof of higher derivatives form Cauchy Integral formula.

pI am studying the Cauchy Integral Formula from Ahlfors' book. I can not
understand some points. If you can discuss them a little, it will be very
much helpful to me./p pI have added this portion from the book. /p pimg
src=http://i.stack.imgur.com/4GpTK.png alt=enter image description here/p
pimg src=http://i.stack.imgur.com/Qc6i6.png alt=enter image description
here/p pimg src=http://i.stack.imgur.com/ZvWhJ.png alt=enter image
description here/p p$$$$$$$$/p ol lipWhy equation (24) is not sufficient
to say that derivative of all orders of the analytic function $f(z)$ are
not analytic ?/p/li lipIn Lemma 3 we are first showing $F_1(z)$ is
continuous. Why? For analyticity we want continuous derivative of the
function $F_1$./p/li lipCan you expand a little the expression $F_n(z) -
f_n(z_0)$ in Lemma 3./p/li lipWhy Lemma 3 is required for a rigorous proof
of equation 23 and 24 ?/p/li /ol pThank you for your kind help. /p