Laurent Series Explained | How to Determine Laurent Series | Complex Analysis #9

Everything you need to know about Laurent Series explained. The video will contain problems on Laurent Series and how to solve them all for each Laurent Series.
The video will include concepts as:
► Definition of Laurent Series
► Principal part and Analytic part of a Laurent Series
► Convergence of the Principal part and Analytic part
► A theorem about when we can use Laurent Series
► How to determine all possible Laurent Series around some point of expansion
► How to solve Laurent Series.
LINK TO COMPLEX ANALYSIS PLAYLIST
• The Complete Guide to ...
LINK TO CANVAS
drive.google.com/file/d/1IHef...
SUPPORT
Consider subscribing, liking or leaving a comment, if you enjoyed the video or if it helped you understand the subject. It really helps me a lot.
CLARIFICATION OF THE METHOD:
To determine all possible Laurent Series:
Start by marking the important points on the graph, the important points are the poles of the function and the point of expansion. The will divide the complex plan into several domains and each of these domains will have its own Laurent Series, since none of these domain will contain any poles of the function.
You may note that most of the time we only need to determine ONE of the parts (analytic or principal part) to determine the Laurent Series. This comes from the fact that both of these parts are only present (converges) if we are inside an annulus domain. If you want to determine a Laurent Series outside of a circle then you only need the principal part (since the analytic part will not converge in this domain) and if you want to determine the Laurent Series inside of a circle then you only need the Analytic part (since the principal part will not converge in this domain).
CONCEPTS FROM THE VIDEO
► Laurent Series
A Laurent Series is a specific kind of power series and is used to approximate an analytic function f(z) around some point z_0 (note that z_0 can be an isolated singularity) in the complex plane
... + (a_-3) *(z-z_0)^-3 + (a_-2) *(z-z_0)^-2 + (a_-1) *(z-z_0)^-1 + a_0 + a_1 *(z-z_0) + a_2 *(z-z_0)^2 + a_3 *(z-z_0)^3 + ...
where the coefficients a_n are determined by contour integration, but 99,9 % of the cases these are determined by geometric series. Note that: "... + (a_-3) *(z-z_0)^-3 + (a_-2) *(z-z_0)^-2" is called the principal part, while "a_0 + a_1 *(z-z_0) + a_2 *(z-z_0)^2 + a_3 *(z-z_0)^3 + ..." is called the analytic part.
►Geometric Series
A geometric series is a series with a constant ratio between successive terms. An example on a geometric series is
1/(1-w) = w^0 + w^1 + w^2 +... for abs(w) smaller than 1
which can be modified to the following
1/(1- 1/w) = (1/w)^0 + (1/w)^1 + (1/w)^2 +... for abs(w) bigger than 1
and the reason these two series are used a lot to solve these kind of problems, is because this geometric series looks pretty much like the Laurent Series we would like to obtain.
► Poles
A pole is a specific kind of singularity, the short and the most intuitive definition is that poles are points z_0 in the complex plane so that f(z_0) = g(z_0)/0, where g(z_0) =\= 0.
► Partial Fraction
The partial fraction decomposition or partial fraction expansion of a rational function is used to express the fraction as a sum of fractions with a simpler denominator.
[Link to video is coming].
TIMESTAMPS
00:00 Intro
00:13 Theorem Laurent Series
01:06 What is an Annulus domain
02:04 Good things to know
03:40 Why geometric series are the best
Examples: Determine all possible Laurent Series for the following functions
05:20 f(z) = 1/(z-2) around z=0
08:02 f(z) = 1/(z-2) around z=1
09:48 f(z) = 1/((z-1)(z-2)) around z=0
SOCIAL
► Follow me on KZbin: bit.ly/1NQhPJ9
► Follow me on Twitter: / the_mathcoach
SOURCES:
Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics: Pearson New International Edition.
mathworld.wolfram.com/LaurentS...
en.wikipedia.org/wiki/Laurent...
sym.lboro.ac.uk/resources/Hand...
zimmer.csufresno.edu/~doreendl...
www.math.ust.hk/~maykwok/cour...
HASHTAGS
#TheMathCoach #ComplexAnalysis #Complex Analysis Playlist

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I have no video covering the subject at the moment. But I can recommend this video (kzbin.info/www/bejne/hnu0o4d5Zs-dm5I) by Barrus, I used to watch him when I took this course myself. If this still is not sufficient, then I recommend that you try to use this (math.stackexchange.com/questions/tagged/complex-analysis) math forum. It is really active, with a lot of helpful members and completely free of course. Hope this helps :)

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[@TheMathCoach]

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Happy you like my teaching methods and I would also like to get the motivation and time to create videos again. I might come back in the future, I at least have to finish the video I started making about 1-1.5 years ago.

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Thank you so much for the kind words and for the tip, I will see if I can implement it in future videos :)

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Good luck mate, I believe you can do it! Hope the playlist will be of use to you, just let me know if you got any questions about the topics I cover in the videos, I will be sure to answer frequently and to the best of my ability :)

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I'm happy to hear that!

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Thank you for the great explanation! One thing I'm curious about is why we did the expansions around 0 and 1; since there is a singularity at 2, wouldn't we want to do the expansion around that point?

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I'm glad that you found it useful :)

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Excellent video - helpful effects, clear handwriting and speech, having the formulae in the corner is very helpful. You could be a little more clear on the fact that some of the series being derived for certain domains are turning out to be only the analytic/principal parts for examples 1) and 2), but small detail. Thank you for posting.

[@TheMathCoach]

I'm happy to help and also thank you for the superb feedback. This is exactly the kind of information I need to improve my content, since this helps me understand what works and what doesn't. Thanks for leaving an comment :)

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No problem, I'm glad you liked it :)

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[@TheMathCoach]

No problem, I got you covered. Glad to see you browsing through some videos, welcome to my channel mate. I have always found it kind of weird why most of the videos online never really explains why we use these geometric series and how to apply them to a specific problem. That is why I thought it would be kind of nice to cover these two parts in depth and I'm glad that this explanation was useful to you!

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[@TheMathCoach]

Ops, forgot to reply here! Hope your contour integration reviewing went well :)

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Any time!

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I'm glad I could help you with your grade :)

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[@TheMathCoach]

Happy to help! Next video coming out this Sunday :)

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[@ethiopiansickness]

Hello Math Coach, thank you for a great video. I just have a question about 7:21. I don't understand when and why you should change the limits of the summation. I've seen this phenomenon occur for other solved problems that I've seen. When should we change the limits of the summation? You changed the lower limit from n = 0 to n = 1. thank you. Also at 5:08 in the top right corner ( small box with formulas) what do the (1^x) and the (2^x) represent ?

[@TheMathCoach]

Hello, happy you liked it. I only changed it here since the *principal part* (which is the part we are working with here in this example) is defined to start from n=1 but our summation started at n=0, therefore we needed to change it otherwise it would the summation could not be the principal part of the Laurent Series since it was not written in the correct form. The symbols (1*) and (2*) is the notation/names I gave the formulas (1* = the first formula, etc.) so it would be more clear in the video which one I'm referring to when doing the examples. Hope it clarified it!

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Thank you sir...

[@TheMathCoach]

Happy to help out and I hope you are having a great day :)

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thanks man......

[@TheMathCoach]

No problem mate.....

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You are reading z_0 as z_NUT 😂 I'm sorry, it just made my day. Thank you for the very helpful explanation of the topic.

[@TheMathCoach]

Happy I could make your day better, and I hope you are having a great day today too, and othervise you could also come back here for a quick laugh :)

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[@TheMathCoach]

Thank you so much 😀

[@wy2714]

At 11:06 when you said add the two terms together, may i know why did you subtract instead?

[@TheMathCoach]

wrong use of words, I inserted the expressions for the fractions back into the formula above.

[@Tiagogo21]

Hello, thanks for the great video ! But something confuses me. for the last example where lzl > 2, you say that for the expansion of 1/z-1 we can use the same expansion as the one in the 1< lzl < 2 domain , I don't understand why ? Normally we must have the expansion of 1/z-1 valid for that last domain lzl>2 but here you write the expansion valid for lzl>1 ? Thanks !!

[@TheMathCoach]

First of I'm happy to hear that you liked the video! In the case with Laurent Series, we manipulate geometric series to get the correct kind of power series which are valid in the right domain. We already determined the principal part for the expression 1/(z-1) (timestamp 12:10) which is only defined where lzl > 1 (according to the formula for the second geometric series). In the last example, we are ones again looking for a principal part for the same expression 1/(z-1) and the only thing that is different from last time is the domain (lzl > 2) which it should be defined in. But we can still use it since lzl > 2 is a subset of lzl > 1 (which implies that the power series is also defined in this domain) and we can, therefore, save time in the last example and use the previously calculated expansion right from the start. I hope this cleared it up, let me know otherwise. Merry Christmas :)

[@Tiagogo21]

Yes !! I was so confused for a moment that I forgot that 2 was bigger than 1 and that it was still valid lol ! Thank you very much for the quick answer. Your videos really help me in my revisions for winter exams! Keep it up and Merry Christmas to you too !@@TheMathCoach

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Hey man, Great content! BTW may I know, what softwares and apps you used to make this video?

[@TheMathCoach]

Hello, check out my channel! I have an video that goes through all the steps :)

[@beatrizgomes8077]

Thank u so much!

[@TheMathCoach]

I'm just happy to help :)

[@Alabdullelah]

WOW!!! You literally made my day . THANK YOU SSSSOOOO MUCH! 8) Just quick question on the second question, why did we have 1

[@TheMathCoach]

Happy to help :) The boundaries you are referring to are corresponding to the circle we drew at 08:10 |z-1| = 1 (radius 1 and centered at z=1. The formula for a circle is |z-z_center| = radius) and if we had used 2 instead of 1 then it would not have been the same circle, the center of the circle would have been the same but not the radius and the circle would not intersect with the pole at z=2 as we want.