Complex Differentiable and Holmorphic Functions (Pt. II)
Point of Post: This is a continuation of this post.
Let be open (we are obviously endowing with the usual topology). We say that a function is differentiable at if there exists a -linear function such that the limit
We call the complex total derivative of at and denote it . Of course, this definition is pure pedantry since any -linear function is of the form for some complex number . Thus, we know that , if it exists, is secretly just multiplication by some constant. We denote this constant and call it the derivative of at . We see then that being differentiable at is equivalent to the statement that
exists, and when it does, it is equal to .
If is differentiable at every point of we say it’s differentiable on . Note then that we get a function defined by . We say that is continuously differentiable if is continuous on . We say that is twice differentiable if is differentiable on , and we define -times differentiable similarly.
The first key theorem about differentiable functions is the following:
Theorem: Let be open. Then, if is such that is complex differentiable at then it is real differentiable at and (i.e. the complex total derivative is equal to the real total derivative).
Proof: Identifying as -vector spaces we see that is naturally an -map which evidently satisfies
from where the conclusion follows by the definition of being real differentiable and being a total derivative.
This allows us to conclude about the complex derivative all the things that were true for real differentiable functions . For example:
Corollary(Chain Rule): Let be open and suppose that and are functions such that is differentiable at and is differentiable at . Then, is differentiable at and .
Proof: By the chain rule for total derivatives and the fact that (complex total derivative equals total derivative) we know that , but this is clearly just the linear map from where the conclusion follows.
Using the same idea we also can conclude the following:
Corollary: Let be open and . Suppose that are differentiable at then:
where we must assume that for the last one.
We also know this fact:
Theorem: Let be open and connected. Then, if as functions on then for some constant . In particular, if then is constant.
Proof: Apply the case for real functions.
Since we know that real differentiable functions are continuous we may also conclude this:
Theorem: Let be open. If is differentiable at then it is continuous at .
 Greene, Robert Everist, and Steven George Krantz. Function Theory of One Complex Variable. Providence: American Mathematical Society, 2006. Print.
 Conway, John B. Functions of One Complex Variable I. New York: Springer-Verlag, 1978. Print.
 Rudin, W. Real and Complex Analysis. New York,NY: McGraw-Hill, 1988. Print.
 Ahlfors, Lars V. Complex Analysis; an Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill, 1966. Print.