Find where a complex function is differentiable

In summary, to determine if a complex function is differentiable, one must check if it satisfies the Cauchy-Riemann equations. The geometric interpretation of complex differentiability is that the function must have a unique tangent line at a point. A complex function can be differentiable at one point and not at others, and the relationship between analyticity and differentiability is that a function is analytic if it is differentiable at every point in its domain. Some common examples of complex functions that are not differentiable include absolute value functions, logarithmic functions, and functions with branch cuts.
  • #1
boogiewithstu
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Using the definition of the derivative find at which points the function f(z) = Im(z)/z conjugate is complex differentiable.
I know that it is not complex differentiable anywhere but I need to show it using the definition and not the Cauchy Riemann equations.
 
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  • #2
Write z= x+ iy, so that [itex]f(z)= f(x+ iy)= \frac{y}{x+ iy}= \frac{xy- iy^2}{x^2+ y^2}[/tex] Apply the definition of "differentiable" to that. (It is sufficent to look at the real and complex parts separately.)
 

1. How do you determine if a complex function is differentiable?

To determine if a complex function is differentiable, you must check if it satisfies the Cauchy-Riemann equations. These equations state that the partial derivatives of the function with respect to x and y must exist and be continuous at a point for the function to be differentiable at that point.

2. What is the geometric interpretation of complex differentiability?

The geometric interpretation of complex differentiability is that the function must have a unique tangent line at a point. This means that the function must be smooth and have no sharp turns or corners at that point.

3. Can a complex function be differentiable at one point and not at others?

Yes, a complex function can be differentiable at one point and not at others. This is because the function may not satisfy the Cauchy-Riemann equations at certain points, but it may still be differentiable at other points where the equations are satisfied.

4. What is the relationship between analyticity and differentiability in complex functions?

Analyticity and differentiability are closely related in complex functions. A function is analytic if it is differentiable at every point in its domain. This means that if a function is analytic, it is also differentiable, but the converse is not always true.

5. What are some common examples of complex functions that are not differentiable?

Some common examples of complex functions that are not differentiable include absolute value functions, logarithmic functions, and functions with branch cuts. These functions have sharp turns or discontinuities at certain points, making them not differentiable at those points.

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