2.2 Complex Differentiation
For and a complex function , is complex differentiable at if the following limit exists, regardless of the direction approaches from:
We can consider to be a bivariate function for . Two main cases we are concerned with are when approaches from the real and imaginary axes:
Expressing as ,
and
By comparing the real and imaginary parts, we obtain necessary conditions for complex differentiability:
By multiplying the second equation by and adding it to the first, we obtain the equivalent form
The identities (2.2.1) and (2.2.2) are known as the Cauchy–Riemann equations.
Although this condition is necessary, it is not sufficient. Consider the function
Let , , and . Then
The derivative along , , or the real axis, vanishes. Along , , or the imaginary axis, it also vanishes. However, the limit is different for any other pair of and , and hence for other directions of approach.
Definition 2.2.1 (Holomorphy).
A function is said to be holomorphic at if it is complex differentiable on a neighborhood of . If is holomorphic for every point in an open connected set , then it is said to be holomorphic over . A function is holomorphic over a compact set if it is holomorphic on a neighborhood of .
Weierstrass provided the following classification:
Definition 2.2.2.
A function is entire if it is holomorphic over .
For the purpose of the following contents, a region or domain will denote a nonempty, open, connected subset of the complex plane.
Theorem 2.2.1.
Let be open, and let be a function. Then is holomorphic on iff and satisfies the Cauchy–Riemann equations.
Proof.
The first part is to prove that any holomorphic function on has continuous first-order partial derivatives in . This requires an argument that will be covered later, specifically in Section 3.2, which states that the complex derivative of any holomorphic function is also holomorphic over the region.
For the second part, let . Assume that and satisfy the Cauchy–Riemann equations at . Let
Then because have continuous partial derivatives, :
where and denotes a value with higher infinitesimal order to , or that . Then letting ,
Taking the limit as , the high order infinitesimals on the right-hand side vanish, and
We will prove later in Section 3.2 that the complex derivative of a holomorphic function is holomorphic. Under this assumption, has continuous second-order partial derivatives, and therefore,
and by the Cauchy–Riemann equations,
and
Adding the equations,
This general type of equation is known as Laplace’s equation, which is a basic example of an elliptic partial differential equation. Define the operator (the Laplacian)
A function satisfying Laplace’s equation is a harmonic function. Thus, the real and complex parts of a holomorphic function are harmonic functions.
Letting , , the Laplacian is equal to:
Proposition 2.2.1.
Let be open and connected and let be holomorphic. It follows that is constant over .
Proof.
Since is real-valued, on . Then by the Cauchy–Riemann equations on , . Similarly, Therefore, is constant.