Where is the function differentiable on the complex domain?

In summary, the poster has a complex function and is trying to determine where it is analytic. They used the Cauchy-Riemann equations and found that the function is only differentiable when x=y. They are questioning whether this is correct and apologizing for their English.
  • #1
asi123
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Homework Statement



Hey guys.
I hope this is the right place to post this question.

http://img641.imageshack.us/img641/70/97822806.jpg

I have this "simple" complex function, and I need to decide where this function is analytic in the complex domain.
So, I used the cauchy riemann equations as you can see, and I got to the condition x=y in order for this function to have a derivation. I know that this function is not analytic anywhere on the complex domain.
Now, according to the solution we got, this function has derivation only in (0,0), is this true, shouldn't it be on all x=y?

Sorry for the bad English.
Thanks a lot.


Homework Equations





The Attempt at a Solution

 
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  • #2
You mean "complex differentiable" here, not "analytic" (the equivalence does not hold unless the function is differentiable on some open set).
 

Related to Where is the function differentiable on the complex domain?

1. What is a complex analytic function?

A complex analytic function is a function of a complex variable that is differentiable at every point in its domain. This means that at every point, the function has a slope or tangent line that can be determined. It also means that the function can be approximated by a polynomial.

2. How do you determine the domain of a complex analytic function?

The domain of a complex analytic function is determined by finding the set of points where the function is differentiable. In other words, the domain is the set of all complex numbers where the function has a well-defined derivative.

3. What is the Cauchy-Riemann equation and its significance in complex analytic functions?

The Cauchy-Riemann equation is a set of necessary and sufficient conditions for a complex function to be analytic. It states that the partial derivative of the function with respect to the real variable must be equal to the partial derivative of the function with respect to the imaginary variable multiplied by the imaginary unit i. This equation is significant because it allows us to determine if a function is analytic and also helps in finding the derivative of a complex function.

4. Can a complex analytic function have a singularity?

Yes, a complex analytic function can have a singularity, which is a point in the complex plane where the function is not differentiable. A common example is the function 1/z, which has a singularity at z=0. However, the function can still be analytic in a region around the singularity.

5. What is the relationship between complex analytic functions and holomorphic functions?

A complex analytic function is also known as a holomorphic function. This means that it is not only differentiable at every point in its domain, but it is also infinitely differentiable. In other words, all complex analytic functions are holomorphic, but not all holomorphic functions are complex analytic.

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