int 0 to infinity (cos(x))/(π^2 - 4x^2): the best thing you'll see next week
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- čas přidán 10. 09. 2024
- Advanced MathWear:
my-store-ef6c0...
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Nice. If you are lazy like me you can also use: cosx/(pi^2-4x^2)=1/4pi (cosx/(pi/2-x)+cosx/(pi/2+x)) now by the subs pi/2-x =u for the first and pi/2+x=u for the second you get: integral 1/4pi (sinu/u) from -inf to inf which is 1/4😂
You literally destroyed this integral in seconds 😂😂😂
@@maths_505 what to do,i'm lazy🤣🤣
My first reflection is also the real analysis method as you described. But I guess the author just wanted to demonstrate complex analysis skill. 😂
2πi times the residue at (say) π/2 is 1/2 (but you are going the wrong way around), so you pick up half the value, which is what you would expect.
I'm loving the graph in the thumbnail! 😍
You often requested me to include graphs so I asked Myers to design the recent thumbnails.
@@maths_505 indeed, I like how it turned out 👍🏾 Next step is to introduce it in the first 30s of the video talking about the area we are calculating to give people a visual picture, then diving into the calculus 👍🏾
i like the new thumbnails style 🔥
4:30: I would save a variable, by letting the radius be 1/epsilon
Very interesting and smart solution. Thank you
Interesting fact, which may or may not have anything to do with this integration. An approximation to cosx known since the 7th century (by Indians) is (pi^2-4x^2)/(pi^2+x^2)
If you divide cosx/(pi^2 - 4x^2) in the integral, the approximation is 1/(pi^2 + x^2) which is an arctan integral, which probably sorts out to 1 in this limit.
Dear Mat505,
I enjoy every bit of your videos.
I am a first class graduate of mathematics and I could tell you are extremely good with integrals
Can you recommend great books to master this?
Regards.
Books won't help as much as the internet especially math stackexchange.
hi i am a bit young but love integrals and stuff like that, i can recommande uyou a books call "(Almost) Impossible Integrals, Sums, and Serie" by Paul J Nahin aand publiched by Springer it's an amazing book with some stuff i had never seen and detailed explication :)
You are literally integrating on a KILROY contour!
I'd love to reference a certain book... But it has become a cliche at this point, I'm loving the new thumbnail designs btw!
Fine I'll reference it, so in gamlin's book there's an entire chapter on the "angular theorem of residues" [not the actual name but too lazy to open the book rn...] which is the same thing as our closed contour residue theorem, the difference being that it's theta*i*the residue at the point. where theta is the change in angle along the path...
I think it's way more detailed in the book...
I searched on Google gamlins book doesn't exist
Leg days are very relaxing, agree
Ah hell nawh bro
It's not the relaxation
It's the satisfaction of having destroyed your legs by working each muscle group to failure and struggling up the stairs 🔥🔥🔥
@@maths_505 Exactly 🔥🔥🔥
Feyman...I(a)=[....e^(-ax)...],con I(0)=I...risulta una eq.differenziale -4I"(a)+π^2I(a)=a/(1+a^2)...omogenea ok,ma la particolare???
For the particular integral, you end up having to use the exponential integral (Ei) function (en.wikipedia.org/wiki/Exponential_integral), so it becomes intractable.
I like leg day too
COMPLEX MAN
But chest is my favorite, sir
I had a killer chest workout on Monday. Gonna hit arms today🔥
Jordan lemma... What are they gonna teach you in that grad school, as it seems like you already know everything and then some?
Jordan's lemma is a standard result taught in undergraduate complex analysis courses - it's nowhere near graduate-level mathematics!
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