Question

"Any branch of the logarithm is analytic with
derivative 1/2​

Answer 1

Answer:

Recall. When you hear “analytic function,” think power series representation!

Definition. If G is an open set in C and f : G → C, then f is differentiable at

point a ∈ G if f

0

(a) = lim

h→0

f(a + h) − f(a)

h

exists.

Proposition III.2.2. If F : G → C is differentiable at a ∈ G, then f is continuous

at a.

Proof. We have

lim

z→a

|f(z) − f(a)| = lim

z→a

|f(z) − f(a)|

|z − a|

|z − a|

= lim

z→a

f(z) − f(a)

z − a

lim

z→a

|z − a|

= |f

0

(a)| · 0 = 0.

Note. The reason for the following definition will become apparent in Theorem

IV.2.8.

Definition. A function f : G → C is analytic if f is continuously differentiable on

G.