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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:


ole.gif


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


ole1.gif


                         ole2.gif


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 ole3.gif 0. In this approach 2) becomes


             ole4.gif


provided the partial derivatives exist.



Approach 2. Δx = 0; Δy ole5.gif 0. In this approach 2) becomes

 

ole6.gif


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


             ole7.gif


or



             ole8.gif



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


ole9.gif

 

where ε1 ole10.gif 0 and η1 ole11.gif 0 as Δx ole12.gif 0 and Δy ole13.gif 0 and the partial derivatives are assumed to be continuous at the point in question.


Similarly, the increment Δv can be written as


ole14.gif


where ε2 ole15.gif 0 and η2 ole16.gif 0 as Δx ole17.gif 0 and Δy ole18.gif 0 and the partial derivatives are assumed to be continuous at the point in question (z0).


Then


ole19.gif  


where ε = ε1 + iε2 ole20.gif 0 and η = η1 + iη2 ole21.gif 0 as Δx ole22.gif 0 and Δy ole23.gif 0.


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



ole24.gif


             ole25.gif


Dividing by Δz = Δx + iΔy and taking the limit as Δz ole26.gif 0 we obtain


             ole27.gif


Thus the derivative exists and is unique.


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