How Do Laurent Series Differ from Taylor Series in Complex Analysis?

[ognik]

My book is a little confusing sometimes, and googling doesn't always help. Just a couple of queries - and please add any of your own 'tips & tricks'...

1) Laurent series (LS) is defined from $ -\infty $, yet all the examples I have seen start from 0 - I can't think of an annulus with a negative radius myself, so do I just use it from 0 and not worry about the negative side of the domain?

2) What is the practical difference between a complex Taylor series (TS) and LS? I have seen suggested that TS is for holomorhpic functions and LS for isolated singularities, but it seems to me those conditions could apply to both TS & LS?

3) A difference I can see is that TS only allows for the region < disk radius, but LS provide for > some R (and also within an annulus) - so for an annulus could we use TS for inside the large radius, LS for outside the smaller radius?

4) For an annulus, couldn't we avoid using LS, Juts take TS of the outer - TS of the inner?

5) Conversely, could we use LS instead of TS, by making the smaller radius 0?
Thanks for all advice.

 

[alyafey22]

My book is a little confusing sometimes, and googling doesn't always help. Just a couple of queries - and please add any of your own 'tips & tricks'...

1) Laurent series (LS) is defined from $ -\infty $, yet all the examples I have seen start from 0 - I can't think of an annulus with a negative radius myself, so do I just use it from 0 and not worry about the negative side of the domain?

There is no relation between the negative radius and the index of the summation. Having a negative sign in the summation means that the function is not holomorphic at least in a circle of some radious.

2) What is the practical difference between a complex Taylor series (TS) and LS? I have seen suggested that TS is for holomorhpic functions and LS for isolated singularities, but it seems to me those conditions could apply to both TS & LS?

All TS are LS but not vice versa. As you said they are used to expand functions with singularities.

3) A difference I can see is that TS only allows for the region < disk radius, but LS provide for > some R (and also within an annulus) - so for an annulus could we use TS for inside the large radius, LS for outside the smaller radius?

By definition of TS it is used to expand the function around a point and the function has to be analytic on that point. So it has no meaning to say that the function has a TS on an annulus.

4) For an annulus, couldn't we avoid using LS, Juts take TS of the outer - TS of the inner?

If the function is holomorphic on an annulus then it has no TS in the inner radius.

5) Conversely, could we use LS instead of TS, by making the smaller radius 0?
Thanks for all advice.

Yes that's way I said that every TS is LS.

 

[math771]

1) Yes, for practical purposes, you can use Laurent series starting from 0 and not worry about the negative side of the domain. This is because most functions that are analytic on an annulus with a negative radius will also be analytic on an annulus with a positive radius, so the Laurent series starting from 0 will still capture the behavior of the function.

2) The practical difference between a complex Taylor series and a Laurent series is that a complex Taylor series is used for functions that are analytic on a disk, while a Laurent series is used for functions that have isolated singularities. This means that a complex Taylor series can be used for a larger range of functions, while a Laurent series is more specific to functions with singularities.

3) Yes, for an annulus, you could use a complex Taylor series for the region inside the larger radius and a Laurent series for the region outside the smaller radius. This is because the complex Taylor series will capture the behavior of the function inside the larger radius, and the Laurent series will capture the behavior of the function outside the smaller radius.

4) Yes, for an annulus, you could also use a complex Taylor series for the outer radius and a complex Taylor series for the inner radius. This would be equivalent to using a Laurent series, but may be more complicated to calculate.

5) Yes, you could use a Laurent series instead of a complex Taylor series by making the smaller radius 0. This would be equivalent to using a complex Taylor series, but may be more complicated to calculate.