# Does f(z) have a singularity at infinity and how can its residue be obtained?

• abode_x
In summary, the conversation is discussing whether the function f(z) = (ze^(iz))/(z^2+a^2) has a singularity at infinity. The person suggests transforming z to 1/w and taking the limit as w approaches 0. If the limit blows up, there is a singularity at infinity. The person also mentions using L'Hopital's rule and approaching the limit along any line on the z-plane, specifically the i-axis. They then ask follow-up questions about getting the residue at infinity, the conclusion in a case where the limit does not exist, and whether the singularity at infinity and its residue are useful in a physical theory.
abode_x
does $$f(z)=\frac{ze^{iz}}{z^2+a^2}$$ have a singularity at infinity?

if so, how do i get the residue there?

Does $$f(w):=f(1/z)$$ has a singularity at w=0 is what you must ask yourself.

Btw - you're really not in the right forum.

Last edited:
okay so i transform z -> 1/w then take lim w-> 0... if it blows up then i do have a singularity... how do i get lim w->0 of exp(i/w) ?

well first, i think i need l'hopitals (for the whole function). then, can i use the fact that when taking a limit it can be approached along any line on the Z-plane? i.e. use the path along i-axis ?

i think the conclusion will be that it blows up. three follow up questions. 1. how do i get the residue at infinity? 2. what is the conclusion in a case wherein the limit does not exist? 3. Is the singularity at infinity and/or its residue useful? (i mean i know the finite singularities are useful in integration, does this arise in some physical theory?)

i am very sorry for posting in the wrong forum.. thanks for all the help

## 1. What is a singularity in mathematics?

A singularity in mathematics is a point on a function where the function is undefined or does not behave as expected. In other words, it is a point where the function becomes infinite or has a discontinuity.

## 2. What are the different types of singularities?

There are three main types of singularities in mathematics: removable, essential, and poles. Removable singularities occur when a function can be redefined at the point of singularity to make it continuous. Essential singularities occur when a function does not have a limit at the point of singularity. Poles occur when a function becomes infinitely large at the point of singularity.

## 3. How are residues used in complex analysis?

In complex analysis, residues are used to evaluate complex integrals. They are calculated by finding the coefficient of the term with the highest negative power in the Laurent series expansion of a function around the singularity. The residue theorem states that the value of a complex integral is equal to the sum of the residues of the singularities inside the contour of integration.

## 4. Can singularities be avoided in mathematical functions?

It is not always possible to avoid singularities in mathematical functions. However, some techniques such as contour integration and analytic continuation can be used to bypass or "go around" singularities to obtain meaningful results.

## 5. Are singularities always undesirable in mathematical functions?

No, not all singularities are undesirable in mathematical functions. In fact, singularities can sometimes provide important insights into the behavior of a function. For example, poles can indicate the location of a possible maximum or minimum of a function.

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