Integral of 1/(x^2+1) from -inf to inf, Contour Integral

[@drpeyam]

Woohoo!!!! 😄

[@hiy9846]

You actually didn't forget the "i" at 14:38 but it got erased at 13:28 haha

[@shandyverdyo7688]

@@hiy9846 it's not. It's from C1

[@tubhyammehta6598]

Pi- m

[@arnavanand8037]

Hoowoo!!!!

[@sparsetable]

Doctor Pi M (Peyam) :0

[@CLeonard484]

Best explanation of contour integral I have seen. Hopefully, y'all continue with a series of more complex integrals.

[@blackpenredpen]

The One and Only Peyam has many more videos on his channel. You can check the link in description. This is meant to be a starter. : )

[@ガアラ-h3h]

Essentially I can be treated as a constant becaus it is one

[@gsniteesh3794]

Please make a video on complex analysis :)

[@whyit487]

Marvel: Infinity War is the most ambitious crossover event! Bprp and Dr. Peyam: Hold my markers

[@blackpenredpen]

Why It? Hahaha thanks!! And also Chester!

[@victorpaesplinio2865]

This video summarizes my calc 4 exam for next month on residue theorem. Thank you very much!

[@Vampianist3]

This is the first time I know that a lot of Peyam’s videos were taken in his office. Always liked the picture beside the board.

[@Erik20766]

Complex analysis is *really* cool but for people to understand it from the ground and then people don’t have to assume it’s magic :) Basically the reason why the residue is important can be explained by the Taylor series of the (analytic, meaning it has a complex derivative) function, breaking it up into a sum of different powers of z. Every term has a nice primitive function [remember integral of x^n=(x^(n+1))/(n+1) ] *except* the 1/z term! (What’s kind of interesting in real analysis is really important in complex analysis here.) It’s the derivative of the complex logarithm which is multi valued. If we integrate on a closed curve, are start and endpoint is the same and so those terms with a primitive function integrate to zero! The exception is 1/z and famously the (complex) logarithm increases by 2*pi*i in one lap around the origin. So the only thing we care about when integrating on a closed curve is where the function has any singularities (else the 1/z term coefficient is necessarily 0 as the function is “nice”) and in those singularities, what the coefficient of 1/z is, and we call that the residue! (Note: some singularities have residue 0, like 1/z^2 in z=0, and do not contribute to the integral. It also does not have to be a simple “pole”, you can also integrate e^(1/z) around z=0 for example. This is an essential singularity where the function approaches infinity and 0 in the same point but it does not matter!)

[@FractalMannequin]

To compute the residue just notice that 1/(z^2+1) = 1/(z-i) * 1/(z+i) and the second factor has no poles at z=i, so its Laurent expansion coincides to its Taylor expansion. The -1-th coefficient of 1/(z^2+1) now must be 1 (i.e., 1/(z-i)'s -1-th coefficient) times the 0-th coefficient of 1/(z+i), which is its value at z=i, namely 1/2i. So the residue is 1/2i (or -i/2).

[@egillandersson1780]

Hokusai's Great Wave and Van Gogh's Starry Night : Peyam is definitely the Artist of mathematics ! Well, about contour integral, I have to work a bit more :-(

[@habouzhaboux9488]

Finally I found a worth watching video about contour integrals

[@tobyinsley9010]

Honestly this is the most intuitive and clear explanation I've found of the process of finding bounds with triangle inequality etc - thanks Dr. Peyam, this was very useful!

[@itachi2011100]

Is it me or is this a lot more interesting than the usual videos? It gives an insight to how you'd work in university.

[@kcldnx3485]

I just had complex analysis last semester and saw the thumbnail. I immediately thought residue theorem and it was super fun to see you use it

[@atrath]

This video brought back some memories from graduate school... I really miss those times...

[@blackpenredpen]

atrath : )))

[@IoT_]

Hello ,Dr. πm) I would like to say thank you that you invited blackpenredpen to your lecture because they asked very useful questions for novices. As for me, I was graduated from the engineering faculty of the russian university and surprisingly I had all these stuff with proofs (surprisingly because even mathematicians have less mathematics than us- engineers ). Anyway , thank you that you reminded me of the best subject - Complex Analysis) . By the way ,Residue in Russian means "Вычет" 😁

[@RapidScience]

Wow this is amazing, he made some amazing points throughout the video

[@cbbuntz]

Peyam's enthusiasm for math is infectious

[@nishan375]

Love dr. P. Very approachable person and loves his subject.

[@manfredwitzany2233]

Calcualtion of the residuum is much easier than demonstrated. As the pole is of first order, you can use the following identity for residuum calculation: \textstyle \operatorname {Res}_{a}f=\lim _{{z ightarrow a}}(z-a)f(z) The term (z-a) cancels out and the remaining function value is equal to the residuum. I have integrated some even more complex functions. This method is pretty easy.

[@chris-hj2qd]

Man, how beautiful is it to see a culture that appreciates learning. So wonderful.

[@kwirny]

Dr Peyam is genius dude

[@GicaKontraglobalismului]

Nice paintings man .... Van Gogh, Hokusai , and something modernist with many colours resembling the Inca Empire flag ....

[@carcaperu4041]

A question about determined integrals that are equal to pi**N As this video shows Integral of 1/(x^2+1) from -inf to inf= pi ?Is there any function integrated on an interval that is equal to pi**2 or pi**N (where N is an integer N=2, 3.....). Obviously the limit of the integral do not contain explicit pi nor the integrated function. Nor the function contains trigonometric (or inverse of) functions.

[@user-wu8yq1rb9t]

One of your best videos in all the time, I watched it for several times! Thank you so much dear *bP🖋️rP🖍️*

[@bandamkaromi]

wow!! Dr. Peyam. Brilliant try. Thank you. BlackPenRedPen!!

[@remlatzargonix1329]

This was cool.....Dr. P. rocks!

[@radiotv624]

This is so wholesome haha I love complex analysis as well!

[@flaviusclaudius7510]

Nasty flashbacks to undergrad ... while complex analysis is amazing, I found it very challenging.

[@douglasstrother6584]

It's more fun without the exams!

[@flaviusclaudius7510]

@@douglasstrother6584 I should probably dig out my lecture notes, then!

[@douglasstrother6584]

@@flaviusclaudius7510 You'll be surprised by what you *did* understand. It's a lot more fun w/o the time pressure.

[@CDChester]

Worth every minute

[@blackpenredpen]

C. D. Chester totallly!!!

[@Pete-Prolly]

Dr. Peyam is stylin!! Look at the sweet watch!! Steady pimpin' these lil Math tricks! Get Σ playa!! (I don't really talk like this; its just fun and I'm just very enthused about your attire.) But yo shirt be lookin' ill af; is dat silk???

[@nimmira]

Ohhh ... Dr Peyam is left-handed!!! :D

[@erikawanner7355]

Where was this channel when I was in college?!?! 😊

[@CofeeAuLait]

MIND BLOWN.

[@ProJeT3Toad]

Uhh Dr.Peyam rocking the Rolex, okay, I see you xD

[@kingbeauregard]

You guys are fun. I want to see a Very Special Episode of "The Big Bang Theory" where you give them all swirlies.

[@CDChester]

i can picture this so vividly

[@birupakhyaroychowdhury974]

Loved it....!!!!

[@lukestanislaus8887]

Really enjoyed this one!

[@PauloYgor1]

Bprp: because "i" don't like to be on the bottom. Dr. P.: Yes, Nice. Bprp: ... Someone don't watch my videos Hahaha hahaha

[@Galileosays]

Great to see this again. We used the limit rule. The Laurent series was for me an eye opener, or should I say an i opener.

[@elizabethmeghana9614]

Wooooowwwwwwwwwwwwwwwwwwwww I really appriciate the teacher , well done sir

[@3ia18_prasetyaharkrisnowo7]

Pls can you make a video about macclaurin series thx.

[@mtaur4113]

Sometimes you do something just to see that it agrees with what you already know. Other times you do it because it is actually easier.

[@DarioCasarotti]

Why didn't you use the evaluation of arctan(x) between inf and -inf? It was much simpler... Did you just want to show an alternative method?

[@blackpenredpen]

yes

[@thfwil]

Happy Pride Peyam!!

[@drpeyam]

Tom FW Happy pride 🙂

[@muhammadqasim7056]

Do the integral of y=x^^x

[@angelmendez-rivera351]

Muhammad Qasim Dilawari It cannot be done. f(x) = x^^x provided f:R -> R is not integrable, since it is discontinuous almost everywhere. In fact, it is defined almost nowhere.

[@muhammadqasim7056]

Ah shit,my bad

[@LouisEmery]

I would think that someone who could do 1000 integrals, would be a complex analysis jock.

[@lordlix6483]

Awesome Video 😄

[@elizabethaugustin5494]

I enjoyed math for the first time

[@Serghey_83]

Res. Integral [C] dF(z)

[@federicopagano6590]

Excelent video dr peyam. I wonder what would have happened ...if when we were about to integrate 1/(z^2+1) wich goes to zero....instead we had put Arctan(z) what would be the result? Again 0? From z=R+0i to z=-R+0i the integrand satisfies Cauchy Riemann so...why not....

[@ThisIsEduardo]

Great video !

[@rot6015]

Chester has a great shirt😄

[@blackpenredpen]

Rot definitely

[@carviryzen288]

Hi Blackpenredpen, I have a question for you: When you write(for example) sin^-1(x), you mean 1/sin(x) or arc sin (x)?

[@luqas8001]

99% sure that its arcus. I struggle with the same problem everyday

[@isaacaguilar5642]

Typically arcsin(x) because we call the other one csc(x)

[@DendrocnideMoroides]

When Dr.Peyam was parameterizing the line segment on the semi circle from (-R,R) he wrote that ( γ(t)=t) but what is the meaning of (γ(t)).

[@MagicGonads]

(gamma(t)) is just enclosing it in brackets so it's easier to square without confusion gamma(t) is just some function of real t that outputs complex values it sorta represents how one might draw the curve in 2D using a pen and t is the time it takes to get to the point gamma(t) while drawing the curve (but the time can be any interval over the reals, it doesn't literally mean real world time since starting the curve, just an analogy) there might be some constraints like maybe it has to be continuous or not undifferentiable at measures greater than 0 or something like that but anyway since it's a real input function it can be differentiated using bounds of integration notation (if it is an integrable function) just like any real functions you would deal with in calculus or real analysis, just that the output space would may complex (in this case since gamma(t) = t which is always real)

[@trafo222]

By the way I think you can do paramatrization this way too. Say z=x+i*y, from -R to R there is only real numbers so z=x if z=x then dz=dx. After that just put in place.

[@deyomash]

I just always forgot why the semi circle is sufficient. Why not both poles have to be included. Because you can integrate using any curve? but you have to include at least 1 pole, clearly.. can't rememberrr

[@elizabethmeghana9614]

I really liked the host

[@BrainsOverGains]

Did you make a mistake? The integral was 1/z^2 + 1 and you used the triangle inequality with 1/z^2-1?

[@dank9427]

1/abs(z^2+1)=1/abs(z^2-(-1)) We needed a subtraction here, and when we broke it up, abs(-1) just became 1

[@bhaveshohal3390]

Man... I love your smile 😍😅😛

[@zacharyusher6577]

"ok" - blackpenredpen 5/1/19

[@blackpenredpen]

Zachary Usher ?

[@Apollorion]

Can't this integral (i.e. from minus infinity to plus infinity of 1/(1+x^2) ) also be solved like this? 1: L=int(-inf,+inf, 1/(x^2+1) ) 1/(x^2+1) is even, so L can also be derived as 2 times the integral from 0 to infinity => 2: L=2*int(0,+inf, 1/(x^2+1) )=2*int(0,+inf, (1/(x+i))*(1/(x-i)) ) 3: f(x)=1/(x^2+1)=1/((x+i)*(x-i))=A/(x+i)+B/(x-i) => A=-B and -2Ai=1 => A=i/2=-B => 4: L=2*int(0,+inf, i/(2(x+i))-i/(2(x-i)) )=i*int(0,+inf, 1/(x+i)-1/(x-i) ) 5: F'(x)=2f(x)/i=1/(x+i)-1/(x-i) and so F(x)=ln|x+i|-ln|x-i|+C L=i*(F(+inf)-F(0) )=i*(lim( x->+inf, ln|(x+i)/(x-i)| ) - (ln(i)-ln(-i)) )=i*(0-i*pi/2-i*pi/2)=-i^2*pi=pi I wonder: how much am I doing (in)correct at step (4 to) 5 or maybe somewhere else?

[@kevincardenas6629]

Isn't it easier to use Cauchy's Integral Formula? with f(z)=(z+i)^-1 and w=i? you get the exact same result. Also this only works for R>1 right?

[@GSHAPIROY]

Nice Big Classroom He Has There

[@alexismandelias]

arctan(inf) - arctan (-inf) = π ?? I'd like an explanation please. You just glossed over that as if it's trivial

[@blackpenredpen]

Alexis Mandelias arctan(inf) is pi/2 And arctan(-inf) is -pi/2

[@alexismandelias]

@@blackpenredpen nvm I'm beyond stupid. I was thinking of a different function. Thanks for reply though

[@RiteshKumar-sy9sp]

Can't we use Euler's form for z??..

[@davidloomis6668]

What course would you learn this in?

[@joeaverage8329]

Integration 101 Just Joking. You would learn this in Complex Analysis usually.

[@hugoschmitter476]

Oke, it’s very interesting

[@hamiltonianpathondodecahed5236]

it flew , just over the cereberum

[@yashovardhandubey5252]

BPRP : "OK" Me : master saitama?

[@francescocipriani8888]

Are you also a doctor? , so do you have phd?

[@غيثالأسعد-ي9ظ]

Wonderful 🌷

[@gtmstev]

Why in that case the absolute value of z its iqual to R?

[@jayapandey2541]

I get it or should I say 2i get it.

[@yoylecake313]

why is it x in the title, not z?

[@yoylecake313]

oh, i commented before watching

[@osamataha9252]

please can help me to solve let a and b are matrices, if a*b=b*a prove that a*(b^-1)=(b^-1)*a

[@bhuvird178]

Super u and Dr πr

[@matefixfix1338]

Okay

[@brandonhh4111]

🏳️‍🌈🏳️‍🌈🏳️‍🌈

[@FaustoTube1972]

I'm the like number 1000........yeeeeeeeeeeeeeeeeeeeeee ;-)

[@dean10956]

I umm... I have no clue what just happened.

[@germangb8752]

3:10 shouldn't it be absolute value abs(γ'(t)) in general?

[@duckymomo7935]

Yes He forgot it

[@VibingMath]

Wow that starry night picture(wrong focus sorry)

[@blackpenredpen]

Mak Vinci oh ok!! Hahaha

[@VibingMath]

@@blackpenredpen Though that wont distract the great lesson by Dr. P 😁

[@blackpenredpen]

Mak Vinci there’s another really cool picture, coming up soon

[@p.singson3910]

Talking about Starry Night, Please make a video chaos theory and turbulence.

[@p.singson3910]

Nice focus Mark, not a worthless one. There's a mathematical/physical backstory to it: accurate depiction of Turbulence.

[@bouteilledargile]

i stan the pride flag in the background

[@user-nb6zu3rk4f]

8:40

[@rurafs7934]

Wait wait... A rainbow flag 🤔🤔

[@angelmendez-rivera351]

Rura FS Ye

[@IanSwart]

Fags use it

[@fuckyou1640]

🌈 Your point is?

[@ummwho8279]

Well yeah, I mean I'm pretty sure Dr. Peyam is.... a wonderful human being and a gift from the heavens, duh. ;)

[@hexeddecimals]

I got a little giddy when I saw that

[@GUTY1729]

Subtitules in spanish please!!!! Tranks.

[@raincold5426]

Oke

[@JamesLaFleur]

Why is there this rainbow flag?

[@fuckyou1640]

Why not the rainbow flag? 🏳️‍🌈🏳️‍🌈🏳️‍🌈🏳️‍🌈

[@sensei9767]

June is pride month

[@JamesLaFleur]

@@sensei9767 Thank you!

[@JamesLaFleur]

@@fuckyou1640 Everyone should live how he wants to. But this is brainwashing. lbpost.com/wp-content/uploads/2017/10/DMN9oKgUIAAls6N.jpg

[@sensei9767]

@@JamesLaFleur 1. This doesn't have to do anything 2. How is education brainwashing? By that logic I could say that it's brainwashing if kids only lern about traditional families and relationships. 3. What's exactly your problem with that picture? You can't really see what those books are about, or is this about the outfit? If you want to have some sort of conversation you need to present your problem in some meaningfull way.

[@anabang1251]

In physics we learn this in 2nd semester lol.

[@blackpenredpen]

Ana Bang 👍

[@blackpenredpen]

We learn this in Peyam’s office.

[@anabang1251]

@@blackpenredpen Serious question: Did you really not know complex integration? You know so damn much about integration and maths in general, it's weird to see you struggle at sth. I can actually do^^

[@blackpenredpen]

Ana Bang I kinda remember it but kinda don’t. I learned this like over 12 years ago and hadn’t touch it since then.

[@sergioh5515]

You learn complex integration in a physics class? 🤔

[@ilouleoy7502]

I get so lonely lonely lonely lonely lonely

[@thephysicistcuber175]

Third

[@GUTY1729]

No entendí una pija pero vi todo el video 📹 por que se cagaban de risa 😂.

[@oscartroncoso2585]

First!

[@DAOXINNo15]

We stan with an LGBTQ+ math professors!

[@High_Priest_Jonko]

Show the teacher some respect

[@blackpenredpen]

Sydney Carton ?

[@dharminderram5246]

What?

[@kaj694]

周知院学院　周知のじーじつ

[@Abdalrhman_Kilesee]

LGBTQ flag ? 😡

[@Biggyweezer69]

Anyone watch while playing minecraft?

[@Biggyweezer69]

@@mipmip4575 same trying to get into 2b2t rn

[@DanielRamos-sl4kk]

Second

[@sinpi6332]

disappointed

[@backyard282]

I was just wondering... Is Dr Peyam gay?

[@luisdaniel9542]

I mean, the " 'i' don't like to be in the bottom" misunderstanding and the huge pride flag might mean something who knows