3.3 The Cauchy-Riemann Equations
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Fix a domain and a function. We saw in Week 1 how to write
where are real-valued functions. Here we will explore what differentiability of as a complex function tells us about the functions and . The result will be the Cauchy-Riemann equations that give two relationships between the partial derivatives of and .
Definition
Fix a real-valued function . Define
for any at which the limits exist. We call and the partial derivatives of .
Thus, to calculate we treat as a constant and differentiate with respect to using the usual rules, and to calculate we treat as a constant and differentiate with respect to . Note that one often sees
written for and respectively.
Example
Calculate and for .
Solution:
When calculating we use the usual rules of differentiation with as the variable, treating as a constant. For we treat as the variable and as the constant. Thus
Theorem (The Cauchy-Riemann Equations)
Let and write . Suppose that is differentiable at . Then the partial derivatives
all exist and the relations
hold.
Proof:
Recall that to calculate we take a limit as . The idea of this proof is to calculate the limit in in two different ways: by taking along the real, and then by taking along the imaginary axis.
Let be real. Taking in the definition of differentiability, we have
because if a limit exists then the limits along the real and imaginary parts exist. Taking next real and we get
by similar reasoning and the fact that . We have two expressions for . Comparing their real and imaginary parts yields the claimed relationships.
The relations in the conclusion of the theorem are called the Cauchy-Riemann equations.
Remark
In particular, we conclude from the proof of the theorem that
wherever is differentiable.
Example
We can use the Cauchy-Riemann equations to examine whether the function is differentiable on . Note that
writing allows us to write . Hence with and . Now
and there are no points at which
holds. We conclude that is not differentiable at any point in .
Notice however that is continuous at every point in . Hence is an example of an everywhere continuous but nowhere differentiable function. Such functions also exist in real analysis, but they are much harder to write down and considerably harder to study. One of the simplest is known as Weierstrass' function
for some and some ; such functions are still of interest in current research.
The Cauchy-Riemann equations hint at what is special about differentiability for a function of a complex variable. Writing again, we can think of as a function . As with any such function, its real derivative at a point is the matrix
provided all partial derivative exist. The derivative of a function can be any matrix
i.e. could be any real number. Compared to this general situation, derivatives of holomorphic functions thought of as functions are very special: the Cauchy-Riemann equations force the structure
on the real derivative.
In particular, if the real derivative of a holomorphic function is non-zero then it is invertible, consisting of a scaling followed by a rotation.
The Cauchy-Riemann equations also tell us that the level curves of and interact in a special way when is holomorphic, as the following theorem explains.
Theorem
Let be holomorphic and fix . Write and . If then the level curves and are orthogonal at .
Proof:
It suffices to prove that the gradients
are orthogonal. But
by the Cauchy-Riemann equations.