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#1 2012-10-31 22:25:37
- zetafunc.
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Laurent Series Help
Find Laurent expansions for:
valid in the annuli;
(a) 0 ≤ |z| < 1,
(b) 1 < |z| < 3,
(c) 3 < |z|.
---
I've found a Laurent expansion but I'm not sure what to do about the different annulus ranges.
which forms the geometric series
or
which I think simplifies to
but I don't know how to address the problem of the annulus ranges given for (a), (b) and (c). I think mine is valid for the range in (a), but I don't know what to do about the others.
#2 2012-11-01 01:44:12
- bobbym
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Re: Laurent Series Help
Hi;
The poles of that are -1 and 3.
Trouble is, when I expand around either one I do not get your principal part.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#3 2012-11-01 01:49:29
- zetafunc.
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Re: Laurent Series Help
I know those are the poles (from the quadratic), but how can I use that here? I thought the principal part was the part of the series involving negative powers -- which part is my principal part here?
#4 2012-11-01 02:02:52
- bobbym
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Re: Laurent Series Help
The principle part is the part involving negative powers. To find the Laurent you expand around one of the poles. Which pole did you use?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#5 2012-11-01 02:16:18
- zetafunc.
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Re: Laurent Series Help
I'm not sure what you mean -- I just got f(z) into a form which I could generate the series of. How do I expand around a pole?
#6 2012-11-01 02:33:57
- bobbym
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Re: Laurent Series Help
Hi;
I just do not see any negative powers in your series. Why is it a Laurent series?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#7 2012-11-01 02:36:31
- zetafunc.
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Re: Laurent Series Help
I don't know. The lecture notes I am looking at say "find Laurent expansions for the function f(z)" and gives the annulus ranges.
#8 2012-11-01 02:52:57
- bobbym
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Re: Laurent Series Help
I can get two Laurent series for that but I do not remember how to handle the annulus stuff. I did not write it down either so I will have to rediscover the method.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#9 2012-11-01 02:56:59
- zetafunc.
- Guest
Re: Laurent Series Help
Go here:
tinyurl. com /bqwts5a
(remove spaces)
and go to Part 6. In there is the exercise I am working on. Unfortunately they have not provided solutions to Parts 5-7, only 1-4.
#10 2012-11-01 03:10:34
- bobbym
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Re: Laurent Series Help
Hi;
They do mention expanding around a singularity.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#12 2012-11-01 03:37:34
- bobbym
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Re: Laurent Series Help
Hi;
p57 and on when they start expanding around z0. This is the pole, the roots of the denominator. You can use the integral or algebraic tricks to get the Laurent series.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.