Need help with finding residue of a simple function

In summary, the conversation is about finding the residue of a "simple" function, specifically f(z) = (1 + z)e^(3/z). The function has an essential singularity at z = 0 and the desired residue is 15/2. The conversation includes the steps taken so far, such as expanding the exponential in a series and understanding the role of Laurent series expansion in finding the residue. The conversation ends with the acknowledgement of needing to study the Laurent series expansion further.
  • #1
joris_pixie
25
0
Need help with finding residue of a "simple" function

Hello,
I'm trying to find the residue z=0 of f(z) = (1 + z)e^(3/z)

I understand this is a essential singularity. I know the answer is 15/2 but I can't seem to find the solution.

I've tried this so far:

f(z) = (1 + z) ( (3/z) + 9/2z² + ...)
But now I'm stuck ! please help :)
 
Mathematics news on Phys.org
  • #2


Do you know what the residue is? You've expanded the exponential in a series - why? It's the correct thing to do, but do you know why?

It's because the residue of a function at [itex]z = 0[/itex] is given by the coefficient of [itex]1/z[/itex] in its Laurent series expansion. So, what's the coefficient of 1/z in your series expansion? (You have to multiply your two series together!)
 
  • #3


Aha !
I'm going to study the Laurent series expansion part again... ;)

Thank you ! :)
 

1. What is a residue in a simple function?

A residue in a simple function is the coefficient of the term with the largest negative power of the variable in the Laurent series expansion of the function.

2. How do I find the residue of a simple function?

To find the residue of a simple function, you can use the formula Res(f, z0) = lim (z -> z0) (z-z0)f(z).

3. Why is the residue of a simple function important?

The residue of a simple function is important because it can provide information about the behavior of the function at a specific point and can be used in the evaluation of complex integrals.

4. Can the residue of a simple function be negative?

Yes, the residue of a simple function can be negative. It depends on the function and the point at which the residue is being evaluated.

5. Are there any special cases where the residue of a simple function is equal to zero?

Yes, there are special cases where the residue of a simple function is equal to zero. This can occur if the function has a removable singularity at the point where the residue is being evaluated, or if the function is analytic at that point.

Similar threads

I Best way to fit three functions
    • General Math
Replies
2
Views
720
I Complex analysis - removable singular points
    • General Math
Replies
5
Views
2K
Finding Residue of Complex Function at Infinity
    • General Math
Replies
3
Views
1K
I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
    • Topology and Analysis
Replies
7
Views
1K
I Solving Trig Integrals with Residue Theorem
    • General Math
Replies
3
Views
955
I Using Residues (Complex Analysis) to compute partial fractions
    • Topology and Analysis
Replies
4
Views
1K
How to find the residue of a complex function
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Compute the residue of a function
    • Calculus and Beyond Homework Help
Replies
2
Views
868
MHB Find Residues for f(z) at $z=-n$
    • General Math
Replies
2
Views
1K
MHB -7.64 Determine the following standard normal (z) curve areas:
    • General Math
Replies
1
Views
771

Hot Threads

Recent Insights

Back
Top