Finding the Residue of f'/f at 0 with a Zero of Multiplicity k

In summary, for the given problem, the residue of f'/f at 0 can be found by first understanding the definition of the residue at 0 and then using it to solve for the residue. The given information suggests that f may be analytic at 0, which can also be taken into consideration.
  • #1
luke1001
3
0

Homework Statement



Let f has a zero of multiplicity k at 0.

Find the residue of f'/f at 0

The Attempt at a Solution



I'm kind of get stuck on this one. I got only this far: Since f has a zero of multiplicity k at 0, then f(z) = (z^k)g(z) :(

Thanks a lot for helping!
 
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  • #2
What is the definition of the residue at 0?
 
  • #3
matt grime said:
What is the definition of the residue at 0?

hmm, do you mean f is analytic at 0?
 
  • #4
You're attempting to find the residue at zero. It would seem like a good starting point to write down the definition(s) of the residue at zero - if you don't know what to find you will never find it.
 

1. What is the definition of residue in this context?

The residue of a function f at a point a is the value of the function f' divided by the function f at that point.

2. What does it mean for a function to have a zero of multiplicity k?

A zero of multiplicity k means that the function has k repeated roots at a specific point. This means that the graph of the function will touch the x-axis at that point k times.

3. How do you calculate the residue of f'/f at 0 with a zero of multiplicity k?

The residue can be calculated by taking the limit as x approaches 0 of the kth derivative of f divided by the kth derivative of f. This can also be written as f^(k)(0)/f^(k)(0).

4. Why is it important to find the residue of f'/f at 0 with a zero of multiplicity k?

Finding the residue can help determine the behavior of a function at a particular point. It can also be used in various mathematical calculations and proofs.

5. Can the residue of f'/f at 0 with a zero of multiplicity k be negative?

Yes, the residue can be negative if the function has a negative derivative at that point. This means that the function is decreasing at that point and the value of the residue will be negative.

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