For a complex function which has an isolated singularity at
1. How do I find out if is a pole or not?
2. If it is a pole what is the order of the pole
Consider for e.g. . Is pole? If yes what is the order of the pole?
Is there a way to find answer to the above question without really writing down the Laurent Series about
A related question I would have is - do we have similar method to comment on essential and removable singularities?
Thanks
Let me just re-phrase (so that I get it)
1. Keep putting higher k (starting with 1), till we get a finite not zero limit for when
Then 'k' is the order of the pole at . Have I got it correct?
Few questions (I tried to work out a logic for this)
1. Isn't this based on the assumption that there is a 'pole' and only thing we need is order?
2. Why is the assumption is analytic in important?
You don't really have to check every one by one: if is more than the order, the limit is 0, and if it is less than the order, the limit doesn't exist. Therefore, if the limit is finite and non-zero, is the right order. In your example, I could have written " , hence hence the order is 2. "
Your questions are related to each other. With your definition of a pole (using Laurent series), you have: "for all in a small disc centered at , and ". This definition concerns not only "at " but also on a neighbourhood of . And it implies that is analytic on a disc around .Few questions (I tried to work out a logic for this)
1. Isn't this based on the assumption that there is a 'pole' and only thing we need is order?
2. Why is the assumption is analytic in important?
The condition , on the other hand, does not say anything about the regularity of outside . If however you know (in addition) that is analytic on a subset (like a disc) around (but deprived of ), then you can see that the function is analytic on the same subset, and has a finite limit at , so that is a removable singularity of (this is a property you probably know). In other words, (with ) is analytic in the previous subset plus the point . Hence for some sequence with . And from the relation , one gets . Thus we find a Laurent series and again the initial definition of pole and order. This justifies my previous post.