Week7Lecture1: Laurent Series

[@XieQiu]

Best series on complex analysis I could find. Very clear, thanks for uploading these.

[@himanshujaiswal98978]

The best lecture I have seen on the Laurent theorem on the internet!!!!! kudos

[@潇厍]

best video I've met!!!! thank you

[@navierstokes2356]

Nice video little Fox.

[@ndapewakalunga7495]

Thank you so much, your video was the best.

[@bassamaboona6444]

Thank you very much for your thorough explanation! I am taking a Theoretical Physics class where it introduces us to a lot of mathematical concepts that have important physical applications, and this video helped a lot! I have been preparing for this course during my winter break! Again, thank you very much!

[@KartyMcFarty]

I know i'm 3 years late but im currently entering my fourth year as a physics major and im curious if laurent series ever comes into play?

[@ClumpypooCP]

Thank you, i was trying to find a good video explaining laurent series. u helped me out!

[@sofiaabrantes4416]

thanks from Brasil!!

[@shawnpheneghan]

f(z)seems to have changed on the slide at 23:30, It started as a product of two fractions and ended as a difference of the same two fractions.

[@emmar1131]

Thank you.I really wish you are my professor.

[@KarlWork-n3i]

In my post when I wrote FFA sorry that was wrong letters l ment FTA which stands for "fundermental theorem of algebra"

[@ditke71]

Excellent lesson, thanks!

[@KarlWork-n3i]

These comments are a bit short just imagine how much time and effort she has put in to make this extremely interesting video. Now look, the field of complex numbers are algebraically closed and as a result a finite degree complex polynomial (that is a polynomial over the field of complex numbers) of degree n can take any complex value what so ever n number of times. P(z) = w where deg(P) = n has n zeros. These n zeros exist thanks to the above the "fundermental theorem of algebra" So the question is WHAT IS A POWER SERIES? What is a Laurent series? Well of course a Laurent series is a power series BUT you can think of it as a POLYNOMIAL OF INFINITE DEGREE!! and ask the question does the fundermental theorem of algebra carry over to analysis and apply to polynomials of infinite degree, that is - Laurent Series? The answer is YES!! So sin(z) has a essential singularity at infinity and can take any value what so ever an infinite number of times. That's because sin(1/z) has a isolated essential singularity at zero. We know that because in ANY neighborhood of it's isolated essential singularity at zero it's LAURENT SERIES is infinite in it's negative powers. It's for this reason that a entire complex analytic function can take ANY value what so ever an infinite number of times with perhaps one exception. For sin(z) there is no exception, a function like exp(z) would miss the one value of zero. That's what Piccard Theorem says it's the generalization of fundermental theorem of algebra applied to ALL entire complex analytic functions. Analytic meaning that they can be represented by a power series at all points in their domain and the series converges to the function F(z) Also an analytic function is represented by a Laurent series with infinitly many negative powers at a point "a" in it's domain IF AND ONLY IF "a" is an isolated essential singularity. (Briefly, in complex analysis a polynomial is an ENTIRE FUNCTION for which the FFA holds. Piccard generalizes FFA to ALL ENTIRE FUNCTIONS!!) This is one of the reasons why complex analysis is so different from real analysis, you don't have FFA in real analysis!

[@egorushakov2267]

Thanks so much

[@zawette]

thanks a lot !

[@محمدحسين-ز8ه2ث]

thank you

[@ElinamSemenyo-pc8tv]

Its not clear oooooo