# Complex Analysis question

by Natura
Tags: analysis, analytic, complex, complex analysis, derivative
 P: 4 Hello, 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 w(z) = u(x,y) +iv(x,y) be an analytic function, where z = x + iy, for some x,y that 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))' = $\frac{∂u(x,y)}{∂x}$ +i$\frac{∂v(x,y)}{∂x}$ So here is where my problem begins. I was doing some problems and then one of them asked me to find $\frac{∂w(z)}{∂z}$, which I believe should be exactly the same thing as the derivative above, but I tried to apply chain rule to it and thus: $\frac{∂w(z)}{∂z}$ = $\frac{∂u(x,y)}{∂x}$$\frac{∂x}{∂z}$ +$\frac{∂u(x,y)}{∂y}$$\frac{∂y}{∂z}$ + i($\frac{∂v(x,y)}{∂x}$$\frac{∂x}{∂z}$ + $\frac{∂v(x,y)}{∂y}$$\frac{∂y}{∂z}$) 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)
P: 1,666
 Quote by Natura Thanks in advance (apologies for my poor Latex use)
What's poor about it? Well, that (w(z))' thing is a little unclear. Would have been more clear to say

$$\frac{dw}{dx}$$

Now, if you did the differentiating correctly, then you should get the same results. So if you don't, then you won't right?

What exactly are all those $\frac{dx}{dz}$ and $\frac{dy}{dz}$?
 P: 597 Is it asking for the Wirtinger derivative? If so, you're actually looking to compute $$\frac{\partial w}{\partial z}=\frac{1}{2}\left(\frac{\partial w}{\partial x}-i\frac{\partial w}{\partial y}\right).$$
P: 4

## Complex Analysis question

Firstly, thank you for the responses.

I agree I wasn't clear enough in my initial post. I'll try to correct that now.

Since
z = x + iy

We can rearrange to get
x = z -iy

therefore
$\frac{∂x}{∂z}$ = $\frac{∂z}{∂z}$ = 1

Similarly for y we get
$\frac{∂y}{∂z}$ = -i

Then using the Cauchy-Riemann relations to eliminate all of the y derivatives and substituting the above results for $\frac{∂x}{∂z}$ and $\frac{∂y}{∂z}$ I get that
$\frac{∂w}{∂z}$ = 2*$\frac{∂w}{∂x}$

As for the Wirtinger derivative, it makes sense the way it is defined but I would like to see how it is derived because I don't see where the factor of (1/2) comes from which is apparently what I am missing.

Thanks again.
 P: 4 Nevermind, I can see that my expressions for $\frac{∂x}{∂z}$ and $\frac{∂y}{∂z}$ are wrong and are off by a factor of (1/2) ... Thanks again.
P: 1,666
 Quote by Natura Nevermind, I can see that my expressions for $\frac{∂x}{∂z}$ and $\frac{∂y}{∂z}$ are wrong and are off by a factor of (1/2) ... Thanks again.
Natura, let me make sure you understand this ok?

We have $w=f(z)=u(x,y)+iv(x,y)$

and:

$$x=\frac{z+\overline{z}}{2}$$
$$y=\frac{z-\overline{z}}{2i}$$

so that:

$$\frac{dx}{dz}=1/2$$
$$\frac{dy}{dz}=\frac{1}{2i}$$

You got that right?
 P: 4 Yeah, I figured it out last time, but thanks for asking. Appreciate it. :)

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