Cauchy-Riemann Equations

Objective of this article: Showing the contrast of the concept of differentiability for real and complex-valued functions.

Assumed knowledge: Basic calculus, properties of complex numbers.

Real differentiability

Definition: A function f is differentiable at x if the following limit exists,

When does a limit not exist?

• When the limit approaching from the right is not equal to the limit approaching from the left (RHL =/= LHL) or,
• When there is a vertical asymptote (i.e., the limit is infinite and the tangent of the graph has infinite slope)

So,

• When the right hand and left hand limits are not equal: that means either the function is not continuous or it’s graph has sharp corners.
• When the limit is infinite or the slope of the tangent at that point is infinite, that means there is a vertical line in the graph or it has a vertical asymptote.

Hence a function that is

• not continuous,
• has sharp edges
• it’s graph consists of a vertical line (or has a vertical asymptote),

the limit does not exist and thus, the function is not differentiable.

Differentiability implies continuity. But, the vice versa is not true. Continuity does not imply differentiability. Hence, the continuity of a function is necessary condition for differentiability, but it isn’t a sufficient condition.

Now, everything above is for functions of one variable. What about the differentiability of functions having multiple variables ? This concept is a part of multivariable calculus and the functions here are said to be ‘real’ differentiable.

In multivariable calculus, continuity of partial derivatives implies differentiability. So essentially, if a function is said to be real differentiable, then it’s partial derivatives exist and it is continuous. (This is a massive simplification of all the proofs and theorems in multivariable calculus that finally lead to this result, but we’ll look into that in a future article.)

Complex differentiability

Definition: Let Ω be an open subset of the complex plane ( C ) and let z be a complex point in the domain Ω. We say that a function f : Ω → C is complex differentiable at z if the following limit exists,

We denote the limit by f ’(z) and call it the derivative of f at z.

In multivariable calculus, the derivative of a real function is a linear map from R^n to R^k. In this case, each point consists of a real and imaginary part which would mean the derivative would be from R^2 to R^2, corresponding to a 2 by 2 matrix and we then obtain a complex number.

Before studying how to reconcile this difference, we look at the consequences of the definition for the real and imaginary part of f. Let’s write h = Δx+iΔy, and f(z) = u(z)+iv(z), which we can also think of as

f(x, y) = u(x, y) + iv(x, y).

So, Re(f) = u(x, y) and Im(f) = v(x, y)

Then the quotient in the definition of complex derivative can be rewritten as,

We could consider multiple ways of sending Δx+iΔy to zero, obtaining the same answer if the limit exists.

We will consider the two obvious options, sending Δx first to zero followed by Δy, and the reverse, Δy first followed by Δx. We find,

This immediately means that at the very least we need to demand some relationships between the partial derivatives of u and v to hold in order to have a complex differential. Namely,

These equations are known as the Cauchy–Riemann equations.

These are clearly necessary conditions, but at this point in no way guarantee that a complex differential would exist if satisfied.

We will now use the Cauchy–Riemann equations and connect complex differentiability with the dependence of the function on the complex conjugate of z (z bar).

Consider f(z) as given by u(x, y)+iv(x, y). Using the fact that,

we can rewrite the function back in terms of z and ¯z. Now, we could consider the derivative of f with respect to ¯z. Applying the chain rule we would obtain,

which we can simplify to,

Notice that if the function is complex differentiable, it satisfies the Cauchy–Riemann equations and therefore the expression above is identically zero. In this sense we say that if a function is complex differentiable, then

The above two equations can also be written as,

Next, we can also define,

Although df/dz makes sense for merely real differentiable f, if f is also complex differentiable at z

then we find that,

so now the Cauchy-Riemann equations can be simply written as,

f being complex differentiable is a lot stronger than f being real differentiable. That is because f being complex differentiable is equivalent to the pair of conditions that f is real differentiable AND that the Cauchy-Riemann equations hold.

Thus, like continuity is a necessary but insufficient condition for real differentiable functions, so is the Cauchy-Riemann equations for complex differentiable functions. It is a necessary condition, but not sufficient to show that the function f is complex differentiable.

MA259 Multivariable Calculus (2020–21), MA244 Analysis III (2020–21) and MA3B8 Complex Analysis (2021–22) lecture notes from the University of Warwick referenced and used for study and research to write this article.