Prove the following theorem:

Theorem. Let w = f(z) = u(x, y) + iv(x, y) be a single-valued function defined in some neighborhood of a point z₀ of the z-plane. Then the derivative f '(z) exists at z₀ if and only if the following conditions are met at the point:

and providing the partial derivatives are continuous in the neighborhood of the point.

Proof.

1] First we prove that a necessary condition for f '(z) to exist is that conditions 1) hold.

In order for f '(z) to exist, the limit

must not only exist but must be the same regardless of the path by which Δz (or Δx and Δy) approaches zero. We consider two possible approaches:

Approach 1. Δy = 0; Δx 0. In this approach 2) becomes

provided the partial derivatives exist.

Approach 2. Δx = 0; Δy 0. In this approach 2) becomes

provided the partial derivatives exist.

Now f '(z) cannot exist unless these two limits a re equal. Thus a necessary condition for f '(z) to exist is

or

2] We now prove that if conditions 1) above do hold then f '(z) will exist. From calculus we know the increment Δu can be written as

where ε₁ 0 and η₁ 0 as Δx 0 and Δy 0 and the partial derivatives are assumed to be continuous at the point in question.

Similarly, the increment Δv can be written as

where ε₂ 0 and η₂ 0 as Δx 0 and Δy 0 and the partial derivatives are assumed to be continuous at the point in question (z₀).

Then

where ε = ε₁ + iε₂ 0 and η = η₁ + iη₂ 0 as Δx 0 and Δy 0.

Employing conditions 1) above, we can write 3) as

Dividing by Δz = Δx + iΔy and taking the limit as Δz 0 we obtain

Thus the derivative exists and is unique.