Determing where function is differentiable (Complex Analysis)

In summary, the function f(x+iy)=1/(x+i3y) does not satisfy the Cauchy-Riemann equations, indicating that it is not differentiable at any point. However, if the equations were to be satisfied, the function would be differentiable at points where x=3y and either x=0 or y=0.
  • #1
scothoward
29
0

Homework Statement


Determine where the function f has a derivative, as a function of a complex variable:

f(x +iy) = 1/(x+i3y)


The Attempt at a Solution



I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?
 
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  • #2
No. Where Cauchy-Riemann is satisfied, the function is differentiable. There could be some places. Are there?
 
  • #3
Hmm I was thinking, maybe I could take the partials of U and V (Cauchy-Riemann) equate them, and find out what x,y have to be in order for the equations to be satisfied. Am I on the right track?
 
  • #4
Yes, if CR is satisfied at a point, then the function is differentiable at that point. That's where CR comes from.
 
Last edited:
  • #5
Alright, so doing the partials and equating I get

x = 3y and
xy = 3xy

But, I am a little confused on what to do from here.

Thanks for the help
 
  • #6
Thinking about it a little more, would this mean when x = 3y, the function is differentiable and when either x=0, y=0, the function is differentiable? Or is my thinking wrong?
 
  • #7
IF that's right then putting the first equation into the second gives 3y^2=9y^2, so y=0. If y=0 then x=0. So differentiable only at 0+0i.
 
  • #8
Thanks for the help Dick!

Just to clarify one more thing. Presuming that I did calculate the paritals right and that leads to 0 + i0, wouldn't that lead to the original function being undefined? As a result, wouldn't the function be differentiable no where?
 
  • #9
Very astute, scothoward. I was on my way to missing that.
 

1. What is the definition of differentiability in complex analysis?

In complex analysis, a function is considered differentiable at a point if it has a derivative at that point. This means that the function must be continuous at that point and its limit as the independent variable approaches that point must exist.

2. How do we determine where a function is differentiable in complex analysis?

To determine where a function is differentiable in complex analysis, we can use the Cauchy-Riemann equations. These equations relate the partial derivatives of a function at a point to its complex derivative at that point. If the partial derivatives satisfy these equations, then the function is differentiable at that point.

3. Can a function be differentiable at some points but not others in complex analysis?

Yes, a function can be differentiable at some points but not others in complex analysis. This is because the Cauchy-Riemann equations involve taking the partial derivatives of the function, and these may not exist at certain points. Additionally, a function may not satisfy the equations at certain points, making it non-differentiable at those points.

4. What happens if a function is not differentiable at a point in complex analysis?

If a function is not differentiable at a point in complex analysis, it means that the function is not smooth at that point. This can have implications for the behavior of the function, as it may not have a well-defined tangent line or slope at that point.

5. Are there any other methods for determining differentiability in complex analysis besides the Cauchy-Riemann equations?

Yes, there are other methods for determining differentiability in complex analysis. These include the use of the Wirtinger derivatives, which are an alternative form of the Cauchy-Riemann equations, and the use of the complex analytic functions, which are functions that satisfy the Cauchy-Riemann equations and are therefore differentiable everywhere in their domain.

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