The Cotangent's Series Expansion Derivation using FOURIER SERIES [ Mittag-Leffler Theorem ]
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- Опубликовано: 24 апр 2019
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Today we are going to derive a very important Pole Expansion, brought to you by Mittag-Leffler! Cot(z) will help us to derive Euler's famous sine product formula! Be prepared for that! =D
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We were watching this in class today and my maths teacher was so confused by all the memes
@@PapaFlammy69 it was the end of the day so we gave up doing work and sir put videos on
We all liked it because we're memelords but my teacher was confused, he loved the product integral boi though
Int x^dx -1
The nipp action moment must've been awkward
14:12 you definitely deserve a big thumbs up for the “ciao” at the end 🔥🇮🇹🇮🇹
Me: *trying to study for final exams*
Papa *publishes a new video*
Me: "Ah shit. Here we go again"
4:39 -t^2(tx) lmao
Quality video as always, Papa Flammy!
I repeated your analysis using sin(t x) on [-\pi, \pi]. After deriving the Fourier series for this function, I let x = \pi/2, which led me to an expression much like the one you got with a bunch of \pi t's everywhere. Letting z = \pi t, like you did, I arrived at another interesting series, namely
sec(z) = 4\pi \sum_{n=1}^\infty (-1)^{n+1} (2n-1)/((2n-1)^2 \pi^2 - 4 z^2).
Wow, in my Fourier analysis class today I just learned how to expand csc^2(x) as a series using Haar wavelets. Are you stalking me?
Flammable Maths i still don’t understand why you split it into sin and cos instead of just using e^ikx. It might just be culture but i think using exponentials is way easier.
Flammable Maths oh it’s because t is part of R\Z since doing trig functions of complex numbers is different, right?
i love how bro is teaching exponentiation rules nowadays
Nice meme! :=)
Incredible content :D
=D
Shouldn't be [-\pi over t, \pi overt t] the interval of integration?? Can We do Fourier series only on the period interval of the function, or on every interval? i dont remeber
the first meme is so true
14:00 make it stop
You have no sense of good taste young one
@@alexismiller2349 I'm not all that young :)
Can go to Berghain together this summer when I move to Europe again. Seem like a hilarious wingman.
I did this literally two days ago for complex analysis!!
@@PapaFlammy69 We used the formula for sin(pi z)/pi z= sum_(n in Z) 1/(z-n)^2 which was super lit... Fourier method is great tho!
3:10
bk≠0
Because t belongs to the The real set of numbers
yeah and b_k would equal to 1/𝜋 integral from -𝜋 to 𝜋 of cos(tx)sin(kx) and cos(tx)sin(kx) would be an odd function only if t and k have the same sign but t∈ℝ and k is defined for k≥1 or k∈ℕ so it isn't confirmed to be odd since t can be negative. Let's just assume the condition that t is strictly positive
13:40 wasn't Mittag-Leffler a name of one mathematician? (I think his first name was Gustav)
0:03 Besso?
You solved the integral the hardest possible way lol. the simplest way is using sum and difference formula
by the way, the closed interval in Fourier series is never valid. At x=+/-L Fourier series converges to the average of the function values at the end points, here cos(t*pi) and cos(-t*pi) are the same, so the average is just cos(t*pi).
That’s pretty imPOR’nt if you ask me :P
That’s cool
=)
um, video end result says sum from 1 to infinity but thumbnail goes from 0 to infinity.. What is it supposed to be? :/ sorry
INTEGERS😘😘😘😘😘 amarlos
n should start at 1 in the thumbnail
Please do a video on why cot(x) is an odd function, but arccot(x) isn't.
I'm not sure arccot(x) is even a function..
arccot(x) is an odd function www.wolframalpha.com/input/?i=arcot(x)
Whenever an odd function has an inverse, that is odd too:
-f(x) = f(-x) → f⁻¹(-f(x)) = f⁻¹(f(-x)) = -x = -f⁻¹(f(x)), and therefore f⁻¹(-f(x)) = -f⁻¹(f(x)) and so f⁻¹ is odd.
Notice the same is never true for even functions: they are necessarily not injective so they never have an inverse function (on a symmetric interval)
On your thumbnail the sigma summation goes from 0 to infinity but your expression goes from 1 to infinity...
Mittag-Leffler is one person!
0:01 mfw b r u h
.... . / .-.. .. -.- . -.. / -- -.-- / -.-. --- -- -- . -. - .-.-.- / .. / .- -- / .. -.-. ... - .- - .. -.-.
Lol fucking legend
Why do we need this expansion?
Ugh. Can't wait. Just give me a little something.
@1:09 otherwise you would be thinking like a physicist my metro-unbearded lumberjack boi
infinity jens gauntlet made burger king vanish :(
0:07 💀💀
Place 0 in cot(x)
Makes Me think I would enjoy attending University in Germany or Austria in English language classes with teachers that have German accents. Pretty blonde froline teachers though.
burger king =0 lol
Dude wtf? Why integration by parts?? See: cos(tx)cos(kx)=1/2cos(x(k+t))+1/2cos(x(k-t))
:v
Pi is not zero😂😂and it already has a proof
Ehh yes it is? 0*0=0 but pi*0=0, so pi=0.
Don't use the F word so lightly, it's poor taste.