# Finding Residue of Complex Function at Infinity

• MartinKitty
In summary, the conversation discusses a problem with finding the residue of a given function using Laurent series. The suggested method is to expand the function and then multiply it with another series to find the desired coefficient. The resulting solution is e^-1 - 1/2 + 1/3!, assuming no mistakes were made.
MartinKitty
Hello everyone,
I have a problem with finding a residue of a function:
$f(z)={\frac{z^3*exp(1/z)}{(1+z)}}$ in infinity.
I tried to present it in Laurent series:
$\frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n}$

I know that residue will be equal to coefficient $a_{-1}$, but i don't know how to find it.

Expand $\frac{z^3}{1+z}=z^3-z^4+z^5-z^6...$. The multiply the two series together to find the coefficient you want (as an infinite series).

mathman said:
Expand $\frac{z^3}{1+z}=z^3-z^4+z^5-z^6...$. The multiply the two series together to find the coefficient you want (as an infinite series).
Then i get:
$\frac{z^3}{1+z}=(-1)^n*z^{n+3}$

and when i multiply I always get ${z^3}$ with some fraction

## \left(z^{3}-z^{4}+z^{5}-\cdots \right)\left(1+\frac{1}{z}+\frac{1}{2z^{2}}+\frac{1}{6z^{3}}+\frac{1}{24z^{4}}+\cdots \right)##, the only interested terms are of the forms ##\frac{a_{-1}}{z}##, that are ##\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}-\cdots ## so it is ## e^{-1}-\frac{1}{2}+\frac{1}{3!} ## (if I did not make mistakes ...)

## 1. What is the meaning of "residue" in complex analysis?

Residue refers to the value that remains after a complex function is expanded into a Laurent series. It is a complex number that represents the coefficient of the term with a negative power in the series.

## 2. Why is it important to find the residue of a complex function at infinity?

Finding the residue at infinity allows us to determine the behavior of a complex function at points that are infinitely far from the origin. This can help us understand the behavior of the function in the entire complex plane.

## 3. How do you find the residue of a complex function at infinity?

To find the residue at infinity, we first need to rewrite the complex function in terms of a new variable, z = 1/w. Then, we can take the limit as z approaches 0 to find the residue at infinity.

## 4. Can the residue of a complex function at infinity be zero?

Yes, it is possible for the residue at infinity to be zero. This occurs when the function is analytic at infinity, meaning it has no singularities or poles at infinity.

## 5. What is the mathematical significance of the residue at infinity?

The residue at infinity plays a crucial role in the Cauchy residue theorem, which states that the sum of the residues of a function inside a closed contour is equal to the integral of the function around the contour. This theorem is widely used in complex analysis to evaluate complex integrals.

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