Complex analysis: having partials is the same as being well defined?

In summary: The solution for one part is pdx and the solution for the other part is qdy. If those two parts are continuous and satisfy the mixed second derivative condition then the function is analytic and the theorem is true.
  • #1
futurebird
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Complex analysis: having partials is the same as being "well defined?"

My professor proved this theorem in class and I don't know if I even wrote it down correctly in my notes. I don't have access to the book so I need to know if this makes sense. Here is the theorem:

Under these conditions:
[tex]R[/tex] is a simply connected region in [tex]\mathbb{C}[/tex] and

p,q : [tex]R \rightarrow \mathbb{R}[/tex] continuous.

[tex]\sigma - rect[/tex] is a rectangle in [tex]\mathcal{R}[/tex].

The existence of a function U on [tex]R[/tex] such that:

[tex]\frac{\partial U}{\partial x}=p, \frac{\partial U}{\partial y}=q[/tex]

[tex]\Longleftrightarrow \displaystyle\oint_{\sigma - rect} pdx + qdy =0[/tex]

---
QUESTIONS:

1. My professor said that having the partials for p and q is the same thing as saying that U is well defined. Why is this the case?

2. Then he said U would be well defined if it was path independent, well, if we know it's path independent then, of course [tex]\displaystyle\oint_{\sigma - rect} pdx + qdy =0[/tex]. So, am I right to think of path independence as a consequence of this proof, not a condition? (I know this seems obvious, but I keep losing track of what we "know" when we start the proof and what we are trying to prove so, I want to get this absolutely clear!)
 
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  • #2
If the partials are continuous, and satisfy
[tex]\frac{\partial p}{\partial y}= \frac{\partial q}{\partial x}[/tex]
, the "mixed second derivative" condition, which for the real and imaginary parts of the function of a complex variable is the same as saying the function is analytic then that is true.
 
  • #3
Okay, I remember that from last semester... That's the Cauchy Rieman condition (or half of it) -- we didn't mention that at all or anything about being analytic--- except I think if U has continuous partials for its real and imaginary parts, it must be analytic... right?

Uggg.. this is so confusing-- If I could get a coherent question together I'd email my professor for help since he has a low tolerance level for hand holding...
 
  • #4
HallsofIvy said:
If the partials are continuous, and satisfy
[tex]\frac{\partial p}{\partial y}= \frac{\partial q}{\partial x}[/tex]
, the "mixed second derivative" condition, which for the real and imaginary parts of the function of a complex variable is the same as saying the function is analytic then that is true.


Are you saying that I left out a condition? Do I need to say that U is analytic in R in addition to the other things I mentioned in the set-up?
 
  • #5
I found out the answer and I thought I'd share. It turns out that the way that we proved the Cauchy's theorem (It's from the book by Alfors) started out by splitting the complex function into two real funtions with the form pdx + qdy.

That's why I only had "half" of the C-R equations.
 

Related to Complex analysis: having partials is the same as being well defined?

1. What does it mean for a function to have partial derivatives?

Having partial derivatives means that the function's rate of change can be calculated in each direction or variable individually, without considering the others. This is useful for studying complex functions with multiple variables.

2. How does having partial derivatives relate to being well defined?

If a function has partial derivatives, it means that it is well defined in all directions or variables. This is because the function's behavior is consistent and does not change depending on the direction or variable being considered.

3. Can a function be well defined without having partial derivatives?

Yes, a function can be well defined without having partial derivatives. This is more common for simpler functions with only one variable, where the concept of partial derivatives does not apply.

4. What are the advantages of using partial derivatives in complex analysis?

Using partial derivatives allows for a more detailed understanding of how a function behaves in different directions or variables. It also enables the use of powerful mathematical tools, such as the chain rule and the gradient.

5. How can we determine if a function is well defined using partial derivatives?

To determine if a function is well defined using partial derivatives, we need to calculate the partial derivatives in all directions or variables and see if they are consistent. If the partial derivatives exist and do not change depending on the direction or variable, then the function is well defined.

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