Another monster integral
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- เผยแพร่เมื่อ 23 ก.พ. 2020
- In this video, I calculate another monster integral, but this time by simply using properties of the fundamental solution of the heat equation. It's surprisingly elegant, watch this video to check out why! Along the way we discover a new special function called the error function.
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Prof: The question on the exam won't be too difficult"
The question on the exam:
You could guide the question! Don't assume they know the error function, give a definition and "some known integrals". It'd make a hell of a problem! (In a Physics degree context lol)
Please prove the Ramanujan's pi formula. I have never found anything on this.
I don’t know what this is
@@drpeyam th-cam.com/video/ZoaEPXEcLFI/w-d-xo.html
See this. And please find out a proof from anywhere.
Yep, it sounds interesting.
@@drpeyam that would be really interesting
@@drpeyam we're #numbertheorygang
awesome video
I never waste my time watching you, son. Go on neverending.👍👍👍
lovely stuff
This make so much sense
Amazing guy in amazing video.
I have been stochastically enlightened. What sucks is I can not prove I actually thought erf before it came up in video. My p-value for the significance of the topics you choose and clarity of your presentation is p = 0.000000005. Wonder where you are going with this heat kernel stuff in general though... I am hooked.
Check out my playlist :)
That was great
Can you make a video together with blackpenredpen and flammable maths, in which you solve hard integrals?
Thanks!
Omg thank you so much, I really appreciate it 😍
So what Maths is trying to say is that :-
If you want to know the future, use me in a right way.
I wonder how these things came out of nothing.
Mathematics describes Physics,
Physics proves that Mathematics is right.
Hey Dr. Peyam can you please tell me (from your PhD. Experience)
Why Mathematics can never go wrong??
Mathematics isnt saying anything because it cannot speak, if anything maybe that the Universe and Mind is incredibly amazing, but you probably dont need mathematics to know that
what makes you think these things came out of nothing?
mathematics is a profound tool, to do physics with
physics cannot prove, only disprove
wrong and right are subtle terms
perhaps something closer to the truth would be a unity
and something bigger than we currently know or understand
Actually these are such type of topics we can't argue upon!!
But (As of I know) Mathematics isn't any tool itself , rather it is a " Tool Maker!! "
It is not used to prove or disprove Physics becoz you don't need to prove what you have,
Mathematics just reaches where our imagination stops!!✌
GREAT!! At 20:09 we can see the following: in the beginning the derivative is a decreasing function, but at some point appears an inflection point. Can we know when this happens?
Taking the second derivative and making it zero, maybe?
@@guitar_jero What if the answer is no? No, there is no inflection point. Are you sure the graph is well done? I prefer an answer from Peyam Tabrizian. See www.math.mcgill.ca/gantumur/math319w11/downloads/slides15.pdf
See www.wolframalpha.com/input/?i=Plot+%281%2F2%29*erf%28x%2Fsqrt%284*80%29%29+and+%281%2F2%29*erf%28x%2Fsqrt%284*8880%29%29+and+%281%2F2%29*erf%28x%2Fsqrt%284*58880%29%29 and www.wolframalpha.com/input/?i=PLOT+%281%2F2%29*erf%28x%2Fsqrt%284*80%29%29+and+%281%2F2%29*erf%28x%2Fsqrt%284*8880%29%29+and+%281%2F2%29*erf%28x%2Fsqrt%284*999958880%29%29
Do you know how to do integral over R of cos(ax)sinc(x+b)sinc(x-b) with respect to x and with a and b two real parameters? The only way I got it was by abusing wolfram alpha Fourier transform.
Have you tried to isolate functions by using the fact that cosx*cosy=1/2(cos(x+y)/2+cos(x-y)/2)?
@@user-fh5rm2ef4n Yes, of course. In case you didn't notice it is sinc and not sin.
@@etienneparcollet727 oh, l'm sorry
@@user-fh5rm2ef4n Don't you fret, it's alright.
If I take this integral on interval ( -infinity;infinity), the result will be 0.,(there are two densilities). I take original integral as(1-integral from minus infinity till 0 from the first part function)-(1-integral from minus infinity till 0 from the second part function). I will get two integrals from minus infinity till 0 . After substitutions r=(y-x)/sqrt(2kt) and s=(y-(-x))/sqrt(2kt) I will get two distribution functions distribution N(0;1). If I am not mistaken, the result is Fí(x/sqrt(2kt))-Fí(-x/sqrt(2kt)),This is declarated in the statistical tables.
Nice
Are you interested in mat statistic? It's been many years, when i studied this branch of mathematics.I remember on cumulative distribution function,in tables are values for cumulative distribution function N(0,1), but error function as a definition in statistic I can not remember.
@@tgx3529 Yes! You should read documentation about erf, its closely related to Fí via complementary functions.
Yes, It's Ok. After the substitution x/sqrt(2) for f(x) for N(0,1) we can get "Peyam's"formula for error function U(infinity,0).On interval (0;infinity), there is symmetry.But I never hear about "error function".Peyam's formula is for distribution N(0;1/2), in statistic we use N(0;1).
Dude, what happened to Blackpendredpen?
Oh, he’s completely fine, just taking a break
Hey! 4:10 Whasssssup!!??
Are you missing a dx from the thumbnail?
No
@@drpeyam my bad for not even watching the vid!!
2nd
But I don't understand for which Kathy is this integral. 🤣
I don't understand what's this guy talking about