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I'm sorry if I'm not posting this to the correct place - this is my first post on PhysicsForums.com

My question regards derivatives of analytic functions. Here it goes:

Let

be an analytic function,w(z) = u(x,y) +iv(x,y)

where

for somez = x + iy,x,ythat are real numbers.

In order to find the derivative of this function, since it is analytic it does not matter from which direction I take the limit in the limiting process so I can easily derive that

(w(z))'= [itex]\frac{∂u(x,y)}{∂x}[/itex] +i[itex]\frac{∂v(x,y)}{∂x}[/itex]

So here is where my problem begins. I was doing some problems and then one of them asked me to find [itex]\frac{∂w(z)}{∂z}[/itex], which I believe should be exactly the same thing as the derivative above, but I tried to apply chain rule to it and thus:

[itex]\frac{∂w(z)}{∂z}[/itex] = [itex]\frac{∂u(x,y)}{∂x}[/itex][itex]\frac{∂x}{∂z}[/itex] +[itex]\frac{∂u(x,y)}{∂y}[/itex][itex]\frac{∂y}{∂z}[/itex] + i([itex]\frac{∂v(x,y)}{∂x}[/itex][itex]\frac{∂x}{∂z}[/itex] + [itex]\frac{∂v(x,y)}{∂y}[/itex][itex]\frac{∂y}{∂z}[/itex])

I get this to equal twice the initially mentioned derivative for all the functions I tried it on.

It seems that differentiating only the real or only the imaginary component (the latter multiplied by i) gives the derivative. I can't explain this to myself. I would be happy if someone points out where my error is.

Thanks in advance (apologies for my poor Latex use)

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# Complex Analysis question

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