Laurent Series for sin(1/z)

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Episode #000054
Monday, January 25th, 2021

One of the simplest examples of a Laurent Series, we will today expand sin(1/z) in the area of the complex plane for absolute values of z greater then zero but less than infinity.
Example taken from page 234 of "Basic Complex Analysis" (3rd Edition) by Jerrold E. Marsden and Michael J. Hoffman.
  1. pentagramprime
  2. pentagramprime
  3. sin(1/z)
  4. Laurent Series
  5. Complex Analysis
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Комментарии
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amazingly well explained video. I always talk myself like this while i am solving math questions lol

kefal
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simple, straightforward, thanks a lot, you saved me

alejandroviciedo
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Love the humor, thanks for the explanation :)

juancanekortegasanchez
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Thansk for the simple explanation:). I was wondering if i could apply this same technique of replacing trig-functions with their taylor expansion for this problem: (cos(z^2)-1)/z^3 at z=0

DkGriefer
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Can you tell me if the domain for the series expansion is 0<|z|<∞

Then the expansion of sin(1/z) is valid in the region |1/z|<∞. So how do we arrive at the solution and prove the domain 0<|z|<∞ is valid ?

gokuvegeta