Is my methodology for checking differentiability and analyticity correct?

In summary, the Cauchy-Riemann equations show that the function f(z) = |z|^2 is only differentiable at z = 0 and its derivative at that point is 0. This means that f(z) is not analytic, as it is only differentiable on the axes and not on any neighborhood around those points.
  • #1
NewtonianAlch
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Homework Statement


State the Cauchy-Riemann equations and use them to show that the function defined by f(z) = |z|^2 is differentiable only at z = 0. Find f′(0). Where is f analytic?

The Attempt at a Solution



f(z) = |z|^2 = (x^2 + y^2)

[itex]\frac{du}{dx}[/itex] = 2x, [itex]\frac{dv}{dy}[/itex] = 0

[itex]\frac{du}{dy}[/itex] = 2y, [itex]\frac{dv}{dx}[/itex] = 0

So x and y must equal 0 for the C.R equations to hold, thus z = 0 + 0 = 0

So this proves they are only differentiable when z = 0.

f'(0) = 2|0| = 0

I'm not sure about the analytic part, f(z) = |z|^2 would be a parabola, so if it's only differentiable when z = 0, i.e. at the origin then it can't be analytic, because an e-neighbourhood at that point will yield points not on the parabolic line, therefore it's not analytic. Is this right or wrong?
 
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  • #2
I just found out that f'(z) = [itex]\frac{du}{dx}[/itex] + i[itex]\frac{dv}{dx}[/itex]

So according to this, f'(z) = 2x + 0 = 2x

Which is 2(Re(z)), not the same thing as what I said earlier about f'(z) = 2|z| = 2[itex]\sqrt{x^2 + y^2}[/itex]

In either case f'(0) = 0
 
  • #3
Hmm, just went over it again and realized, that if it's differentiable only when x = 0, and y = 0, then it's only differentiable when it's on the axes, and hence not analytic because an e-neighbourhood of any size on the axis is a point outside the axis, and hence not differentiable.
 

FAQ: Is my methodology for checking differentiability and analyticity correct?

1. How can I determine if my methodology for checking differentiability and analyticity is correct?

There are several ways to determine if your methodology is correct. One way is to compare your results with those obtained from other established methods. Another way is to thoroughly review and validate each step of your methodology to ensure it aligns with established principles and techniques.

2. What are some common mistakes to avoid when checking differentiability and analyticity?

Some common mistakes to avoid include assuming differentiability based on visual inspection alone, not considering all possible cases, and not properly defining the function or domain of interest. It is important to carefully review and validate each step of your methodology to avoid these errors.

3. Can I use a computer program to check for differentiability and analyticity?

Yes, there are many computer programs available that can assist with checking for differentiability and analyticity. However, it is important to understand the underlying principles and assumptions of the program and to carefully review the results to ensure they align with your expectations.

4. Is it necessary to check for both differentiability and analyticity?

Yes, it is important to check for both differentiability and analyticity because they are distinct concepts. A function can be differentiable but not analytic, or vice versa. Therefore, it is important to evaluate each separately to fully understand the properties of the function.

5. What should I do if I am unsure about the correctness of my results?

If you are unsure about the correctness of your results, it is a good idea to seek feedback from other experts in the field or consult additional resources to verify your methodology. It may also be helpful to go back and review each step of your methodology to identify any potential errors or oversights.

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