Let's do complex analysis: Integrating sin^2(x)/x^2(x^2+1) from 0 to infinity using residue theorem
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- Опубликовано: 14 окт 2024
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Let us tackle this integral once again! We are going to combine the tools of complex and real analysis, featuring papa cauchy's residue funhouse, feynman integration and contour integration! Have fun =)
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Your use of the phrase "curvy boi" alone made me subscribe.
bro i love the way you start almost every integral in I(t) then diff. it just love this way to solve problems😎😎
mihir chawla Me too, the leibniz rule is epic af.
Gotta love Jordanian's Lemon sometimes
How do you even come up with these things... just wow. Keep up the amazing work!
This one was fucking hard! It's not just calculus. It's real and complex analysis
At this point I just think of it as
🌈 math🌈
Thanks for saving me for my complex analysis final, I couldn't have done it without these videos
Flammable math,Can you please solve the integral x/(e^x-1) from 0 to infinity using complex analysis?The answer is equal to π²/6
As a shortcut for the epsilon contour avoiding the origin, you could use the residue theorem for it, then multiply it by the fraction of the circular revolution around the pole at the origin (-1/2). It's negative due to the negative orientation of the epsilon contour around the pole at the origin. You also arrive at -pi*i.
Ok, you love the Feynman technique, no doubt. But it's an overkill here, really. Just split the sin(x)^2 in e^(2ix) - 1 and e^(-2ix) -1 both times a factor -1/4. This way the pole at x=0 is simple. Then integrate over your Gamma/Lambda/real axis countour in the upper half plane for the first part and over a similar, specular, Gamma/Lambda/real axis contour in the lower half plane for the second part and after some book keeping, taking into account the residues at + / - i and half the residue at x=0, you're done. Well, nearly so,. To write it up would take me half a page, definitely not more. Anyway, I have fun watching your video's. Cheers!
Theo Hermsen that is what I thought of too :)
Because math wizards prefer doing it the most complex way same with programmers, this way it makes them appear more superior, because they know the majority doesn't know the language, so they get away with it...
I actually think his way is more natural because
making (sin(x) → sin(tx))(cos(x) → cos(tx))(e^x → e^(tx)) is very common
a quick way to deal with the x^2 term: first use partial fraction = int ( sin(x)^2/x^2 ) - int (sin(x)^2/(x^2+1)) for the second term, use double angle formula then use residue theorem. Next, for the first term int ( sin(x)^2/x^2 ), we use integral by part, int ( sin(x)^2/x^2 ) dx = - int (sin(x)^2) d (1/x) = - sin(x)^2/x + int (1/x) d sin(x)^2 the first part evaluate at 0 and inf give both zero, the second part int (1/x) d sin(x)^2 =int (sin(2x)/x) dx, use u-sub, let u=2x, and use the fact int (sin(u)/u) du = pi/2, we are done!
your vids are great, I N F I N I T Y B O I
You should try solving the more general integral where the squared terms are raised to the n, see what happens
You took the limit of the three integrals on the right handside; that was with epsilon approaching 0, and with R approaching infinity... however, you have to maintain the equality at both sides, you should have done the same thing to the left hand side by getting the limit of negative i(pi)/e^2t as epsilon approaches 0 and as R approaches infinity.
I was confused, but let me guess; the reason you didn't show the solution is because 't' is independent of R & epsilon, and since taking the limit of the constant won't change anything because the limit of a constant function is still the constant itself. Well I'm not sure if my assumption was right, but anyway your videos are awesome and I love them.
also, when you create the functional I(t), and then look at where it's derivative vanishes to come up with new bounds for integration, i can't help but be reminded of homotopy. really neat!
At around 14:00 I see that you make this substitution from z->phi but I thought when you're integrating along the contour that's curved as such, you had to parameterize the curve in exactly the same way you did. It seemed like you implied you made that substitution out of nowhere. ALSO isn't that just 2pi* i residue at z=0. Which is a simple pole, even thought it looks like a second order one.
Great Papa .... I love it
Thank you so much dear lovely Papa ❤️
You should do more complex analysis videos. Complex analysis = nice warm chocolate chip cookie.
:D
Great video! :)
It’s been a while I’ve done complex analysis.
Or you could make the substitution x^2 -> x^2 + a^2 in the denominator. You then have poles at z = +- i and +- ia, and then take the limit a -> 0 after calculating the integral.
what's the short audio clip you play at the beginning of all these videos?
btw: you've totally inspired me to enjoy integration lately! Your videos are awesome!
You saved me hours of pain trying to figure out problems similar to this.
I differentiated a second time which gets rid of the pole at the origin and makes it a lot simpler. Integrating the result twice then gives the same answer
FUZE TEA at the beginning i regonize it!! xD I always drink that at school! :D
nice work! you do not need to change the sign of the second term to apply the backwards triangle inequality, it holds regardless.
Lamda is λ οr the capital Λ
And epsilon ε or the capital Ε
Great videos!
Respect from greece!
Beautiful 😯😯
Can you please do this one??
Integral from 0 to 2pi exp(cosx)dx
Can you make video on basics of contour integration ,I did not understand that how you are going to select contur as a semicircal in upper half
GREAT! Fabulous explanation. Viele Grüße aus Pomerode - Santa Catarina - Brasilien
@@PapaFlammy69, what is the meaning of "boi"?
what is the name of that amazing music when you show the problem!!!????
o wow! this is your second channel!! fapmid is a great writer by the way i absolutely love him!!
hi,i wanted to asked you something,i can´t decide between studiying math or physics (at college level)
is there any piece of advise you could give me?
by the way, i love proving theorems in math,do you do proofs in physics too?(not experiments),because in highschool we didn´t and to be honest i dont know anything about college physics
thanks a lot!
intro with sipp = instalike
Moar complex analysis PLS! Could you also do quantum physics and hard differential equations?
Complex analysis is love
I really like your video, you are very good in explaining also little details one's usually forget. But I have to ask you another integral that is driving me crazy: Integral from 0 to Infinity of Sin^2 (Pi x)/(1-x^2)^2. I tryed with the Feynman trick as you did, but eventually i cannot eliminate the term in the denominator that make me diverge this integral when epsilon goes to 0. Do you have a tip for me?
Can you please solve a tough question of quantum mechanics
Hey swastik ur doing Bsc
Yay, complex analysis
Please solve limit 0 to infinity integration sinx/x(1+x^2)
Answer is. π/2 (1-1/e)
you are the best
Why you choose e^(2itz) instead (e^iz-e^-iz)/2 ?? You always choose e^(itz) if you have sin(x) or cos(x) in your integrand ?
Wow... this was amazing, but there was one thing i didnt get. At 9:00 you said that the contour integral of f(z) would be equal to 2pi*i*residue. Is that some theorem i havent learned yet or am I just to stupid.
Thanks for the help in advance =)
My birthday gift
Thanks 😘
my daily dose of math
15:42 it doesn’t seem valid to take the limit as epsilon goes to 0. I feel that we may need a convergence theorem or perhaps an estimation.
Great work keep it up :)
amazing video !!
Question! domain of the integral is fm 0 to infinity so, the final result isn't half value of your calculations?
Did you use Stein and shakarchi for learning complex analysis
this helped soooo much!! thank you
Great as always :D
Reverse triangle inequality
And they just say make it zero ,now it makes sense
Are you able to do this by differentiating wrt t twice to completely get rid of the x^2 term? I just tried doing this on my own but I couldn’t get the right answer, but admittedly I don’t really know complex analysis
Could you do a video on 1/3 factorial with use of the gamma function?
Very awesome
take a shot for every squiggly boi
Great!!!
So, amazing have a cheers🍻.:D
mil gracias
can you find the center of mass of the area given by the integral of any continuous function f(x)?
when you say poles above the real axis, does that count for z=0 also?(origin)
😩this is soooo cooolll😍
Complex analysis is hard
Nope it ia intresting
@@surjeetsingh1729iitk Something can be both interesting and hard. I'm currently trying to teach myself complex analysis before uni and its really fun, but hard.
Why can you switch sin(2tx) with e^(2i*t*z)?
It's clearly not the same. Or am I missing something here?
You set f(z)=e^2itz/(z(z^2+1))
How did you know to choose that function?
@@PapaFlammy69 Thanks!!
Awesome!
How do I know if the real integral is equivalent to the contour I've constructed?
It's not. The contour contains the original integral, and then some other parts. That's why it is so useful; you generally get expressions for all the parts in terms of your original integral, or as constants, and then an expression for the entire contour integral as a constant, either due to residue theorem or due to integral theorem, which is basically a special case (Residue theorem: A contour integral over a simply closed curve containing pointwise singularities of the function will be equal to the sum of the residues of the function at these singularities. Integral theorem: A contour integral over a simply closed curve containing *no* pointwise singularities of the function will be equal to zero.) Then the rest is algebra.
cant solve this using direct residues; plz make a video for solving this using direct residue
WE WANT BRANCH CUTS WE WANT WINDING INTEGRAL!
When boys do complex analysis
I thought i knew some math... what a fool am I. This was completely out of my league
integrate (e ^ (- s * x) * senx)/x dx from 0 to ∞
If u have differentiated once differentiating twice won't do anything bad but better and then integrating twice.
What is the name of the Lemma?
Whoever asked for the Jordan's lemma proof wants to apologise so bad 😂😂😂
Ur website is not accesibly can u give me the correct lien 😘😘
Very nice, curvy boi!
That's so cringy yet so hilarious. Whooph!
Good video by the way, as always
brilliant
:)
Why quantum mechanics is so hard
Well, is it really?It's all about very hard maths and because we are not able to solve it, we approximate it.
new song???
can't you just complexify the whole thing and do partial fractions?
In the title you should say "integrating" instead of "doing the integral" !! Love your videos btw :)
no. certainly not.
8:38 what, where that come from?
:o
Flammable Maths Sir, is this method contour and residue thereom learnt at master level in university ?? Thought I am math major at at level of bachelor and I didn't see this method yet ! Sir plz reply
Are you still at university? Complex Analysis should be offered at undergraduate level.
It's the easiest one !
25:00
Shit just got real.
OOf spicy af
Yo flammy i certainly follow how you are doing this but on my own i would never get this in a million years...
How long after starting calculus is this actually
Rwesidou
gosh. cant stop thinking why this guy can be so handsome #(facepalm).
Yay 10th comment :)
What a shit entry 0:00
xD
same spiel as here hahahaha
It's actually a bit sad that people spend a large chunk of their lives doing this.
kaanp kahe raha hai be
I gave you a dislike
thx