2.2.1 Wirtinger Derivatives
We have previously introduced the concept of expressing a complex function as a function of and . It can also be expressed in terms of and , where and . Then , , and . By the rules of the derivative, it is only natural that we define
and
If (2.2.4) is set equal to , then it is the equivalent form of the homogeneous Cauchy–Riemann Equations. Then for a holomorphic function , the Wirtinger derivative .
In terms of and , the two derivatives of a function are equal to:
and
If is holomorphic,
On the contrary, by the rules of the derivative,
and
The Laplacian is equal to
Under this definition, we can derive the chain rule:
Theorem 2.2.2 (Chain Rule).
Let be a region such that and . Writing , it follows that
Proof.
Write . Let
so that with . Let be regarded as a function of the real variables ; equivalently we may view as where . The composition is .
Using the real chain rule (provided by the continuous differentiability), we have
Hence,
Now recall
Thus,
Then by substitution,
The terms in brackets equal and . Thus,
A similar calculation using (2.2.5) gives
These are exactly the proclaimed identities.
Last we have taking derivatives of conjugates:
Theorem 2.2.3.
Let where is a region. Then
Proof.
Write and with . Then . We compute
On the other hand,
Taking complex conjugates yields
Similarly,
while
Taking complex conjugates gives