Finding Derivatives of Analytic Functions: Chain Rule Confusion

In summary, the conversation discusses the calculation of derivatives for analytic functions and the confusion surrounding differentiating only the real or imaginary components. The use of the Cauchy-Riemann relations and the Wirtinger derivative is also mentioned. The conversation concludes with a clarification of the correct expressions for \frac{∂x}{∂z} and \frac{∂y}{∂z}.
  • #1
Natura
4
0
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))' = [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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  • #2
Natura said:
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

[tex]\frac{dw}{dx}[/tex]

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 [itex]\frac{dx}{dz}[/itex] and [itex]\frac{dy}{dz}[/itex]?
 
  • #3
Is it asking for the Wirtinger derivative? If so, you're actually looking to compute $$\frac{\partial w}{\partial z}=\frac{1}{2}\left(\frac{\partial w}{\partial x}-i\frac{\partial w}{\partial y}\right).$$
 
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  • #4
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
[itex]\frac{∂x}{∂z}[/itex] = [itex]\frac{∂z}{∂z}[/itex] = 1​
Similarly for y we get
[itex]\frac{∂y}{∂z}[/itex] = -i​

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

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.
 
  • #5
Nevermind, I can see that my expressions for [itex]\frac{∂x}{∂z}[/itex] and [itex]\frac{∂y}{∂z}[/itex] are wrong and are off by a factor of (1/2) ... Thanks again.
 
  • #6
Natura said:
Nevermind, I can see that my expressions for [itex]\frac{∂x}{∂z}[/itex] and [itex]\frac{∂y}{∂z}[/itex] are wrong and are off by a factor of (1/2) ... Thanks again.

Natura, let me make sure you understand this ok?

We have [itex]w=f(z)=u(x,y)+iv(x,y)[/itex]

and:

[tex]x=\frac{z+\overline{z}}{2}[/tex]
[tex]y=\frac{z-\overline{z}}{2i}[/tex]

so that:

[tex]\frac{dx}{dz}=1/2[/tex]
[tex]\frac{dy}{dz}=\frac{1}{2i}[/tex]

You got that right?
 
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  • #7
Yeah, I figured it out last time, but thanks for asking. Appreciate it. :)
 

1. What is Complex Analysis?

Complex analysis is a branch of mathematics that studies functions of complex numbers. It deals with properties and behavior of complex-valued functions, as well as their derivatives and integrals. It is useful in many areas of science and engineering, including physics, chemistry, and engineering.

2. What are complex numbers?

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. These numbers have both a real part (a) and an imaginary part (bi) and can be graphed on a complex plane.

3. What are some applications of Complex Analysis?

Complex analysis has many applications in various fields such as signal processing, fluid dynamics, quantum mechanics, electromagnetism, and more. It is used to solve problems involving electric and magnetic fields, fluid flow, and heat transfer, among others.

4. What is the difference between real analysis and complex analysis?

Real analysis deals with functions of real numbers, while complex analysis deals with functions of complex numbers. Complex analysis also has some additional tools and techniques, such as Cauchy's integral theorem and the Cauchy-Riemann equations, which are not present in real analysis.

5. What are some important theorems in Complex Analysis?

Some important theorems in Complex Analysis include the Cauchy's integral theorem, Cauchy's integral formula, and the Cauchy-Riemann equations. Other important concepts include Laurent series, residue theorem, and the maximum modulus principle.

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