Order of poles: 3 Find Pole and Order for f(z): Residues Complex Homework

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In summary, the problem is asking to find the pole and its order for the function f(z) = \frac{z^3}{(2z+1)^3}. The solution involves finding the poles at z = -1/2, e^(iπ)/2, and e^(-iπ)/2. However, e^(iπ) and e^(-iπ) are not poles since they both equal -1. The correct approach is to use the formula for poles in a function of the form f(z) = \frac{g(z)}{(z-a)^n}, where "a" is the pole with order n. The residue at -1/2 is found by using the formula res at -
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Homework Statement


find the pole and it's order for f(z) = [tex]\frac{z^3}{(2z+1)^3}[/tex]


Homework Equations


[tex]\frac{1}{(2z+1)^3}[/tex] as [tex]\frac{1}{8(z+1/2)^3}[/tex]


The Attempt at a Solution



poles at z = -1/2, [tex]\frac{e^(i*\pi)}{2}[/tex], and [tex]\frac{e^(-i*\pi)}{2}[/tex]
 
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  • #2
e^(iπ) and e^(-iπ) are both -1, so they're not poles. I think you're overthinking the problem. For any function [tex]f(z) = \frac{g(z)}{(z-a)^n}[/tex], "a" is a pole with order n. If the denominator was (z-a)^m*(z-b)^n, then both a and b would be poles, with orders m and n.
 
  • #3
Ok, I know -1/2 had to be a pole but I meant e^(ipi)/2 which is also -1/2
and also when i try to find the residue at that point I end up getting constant/zero...

ie res at -1/2 = (-1/2)^3/24(-1/2+1/2)^2 is that wrong?
 

FAQ: Order of poles: 3 Find Pole and Order for f(z): Residues Complex Homework

1. What is the order of poles?

The order of poles refers to the number of times a function has a singularity at a certain point. In other words, it is the highest exponent of the denominator of the function's Laurent series expansion at that point.

2. How do you find poles of a complex function?

To find poles of a complex function, you need to factor the denominator of the function and solve for the values that make the denominator equal to zero. These values are the poles of the function.

3. What is the order of a function at a pole?

The order of a function at a pole is equal to the number of times the function approaches infinity or becomes undefined at that pole. It is also equal to the number of terms in the Laurent series expansion of the function at that pole.

4. How do you find residues of a complex function?

To find the residue of a complex function at a pole, you need to evaluate the function at that pole and multiply it by the corresponding exponent of the pole. This value is the residue of the function at that pole.

5. How are residues used in complex analysis?

Residues are used in complex analysis to calculate integrals of complex functions. They allow us to evaluate integrals without having to solve complex integrals directly, making complex analysis much more efficient and practical.

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