Complex Analysis/Complex differentiable, holomorphic, Cauchy–Riemann equations

Open sets in the complex planeEdit

Since the complex numbers   are nothing but   with a multiplicative structure, the notion of open sets carries over from   to  . Hence, we consider a set   open if and only if   is open as a subset of   (that is, around each point   there exists a ball around   completely contained within  ) w.r.t. Euclidean norm (or any other norm, due to norm equivalence).

Complex differentiabilityEdit

Recall that a function   is differentiable at   iff the limit

 

exists, and in this case the derivative   is defined as the value of that limit. By analogy, we infer the analogous definition for functions  :

Definition 2.1:

Let   be open, and let   be a function. Let  . We say that   is complex differentiable in   iff

 ,

where this time   is complex, exists, and in this case we write   as an abbreviation for that limit. If the function is complex differentiable on all its domain of definition (in this case  ), we say that   is holomorphic.

The derivative is linear in the following sense:

Theorem 2.3:

Since a complex number is a tuple   in  , a map   is tantamount to a map  . The complex differentiability of   at a certain point implies its real differentiability, in the sense that the directional derivatives exist. In fact, we will later prove that in case of holomorphy, even continuous differentiability of   (in the sense of existence of partial derivatives) will follow, and hence, we have a Jacobian matrix which equals the differential of  . However, the converse is not true: If   in the sense of real numbers (that is, considering   as a map  : All partial derivatives exist and are continuous), we don't know yet whether   is holomorphic or not. The next section will make that precise.

The Cauchy–Riemann equationsEdit

If   in the real sense, then there is a precise criterion for when   is complex differentiable. This is given by the Cauchy–Riemann equations:

Theorem 2.3:

Let   be continuously differentiable at  , that is, all partial derivatives exist and are continuous, both at  . Then   as a function   is complex differentiable in   if and only if the Cauchy–Riemann equations, given by

  and  ,

are satisfied, where  .

Proof:

For the whole proof, note that if  , then   for  .

If   is continuously differentiable in  , then

 ,  ,

where   means  . Thus, in this case,

 

If the Cauchy–Riemann equations are satisfied, we may replace   by   and   by   in the latter expression and obtain

 

(whereby we also obtained another formula for the complex derivative). On the other hand, if the limit

 

does exist, then in particular we are free to choose   for real positive   to get

 

and similarly   for real, positive   to get

 

and hence the Cauchy–Riemann equations. 

Rules for the complex derivativeEdit

For the usual real derivative, there are several rules such as the product rule, the chain rule, the quotient rule and the inverse rule. Fortunately, these carry over verbatim to the complex derivative, and even the proofs remain the same (although we will repeat them for the sake of completeness).

Theorem 2.4 (chain rule):

Let   be complex differentiable at   and let   be complex differentiable at   (this means of course that   must lay in  ). Then   is complex differentiable at   and

 .

Proof:

Set

 

Then

 

by the continuity of   at   (which can be easily proven by multiplying the limit definition of complex differentiability by   and observing that the limit is then   by multiplicativity of limits). 

Theorem 2.5 (product rule):

Let   be complex differentiable at  . Then the product function   is complex differentiable at   and

 .

Proof:

  

Theorem 2.6 (quotient rule):

Assume that   are complex differentiable at   and  . By continuity of  , the function   exists on a small ball around  . It is then complex differentiable at   and the derivative equals

 .

Proof:

The derivative of the function   is given by  ; for

 

Hence, product and chain rule imply

  

Theorem 2.7 (inverse rule):

Assume   is a bijective function that is complex differentiable at  . Then   is complex differentiable at   and

.

Proof:

Derivatives of polynomialsEdit

Theorem 2.8:

Let  . Then a derivative of   at any   is given by

 .

Proof:

 

by the binomial theorem. 

Due to the linearity of the complex derivative, we can now compute the complex derivative of any polynomial, even with complex coefficients:

 .

ExercisesEdit

  1. Compute the complex derivative of the polynomial  .