A ridiculously awesome integral with an epic result

Here's a golden integral with a fascinating solution development linked to the gaussian integral. The Laplace transform made the evaluation all the more epic!

[rajendramisir]

A reliable heterogeneous mixture that leads to a succulent solution of this esoteric integral. Clever choice of tools, techniques and clear explanation. Thanks for sharing. Recognizing and applying identities is a very helpful technique to solve problems in Mathematics.

[omaranbar]

It was a good plot twist at the very end.
Golden Ratio: have you forgotten me boys?

[fartoxedm]

I just discovered your channel yesterday, but I'm already in love! You remind me of my teacher, who always left out some details, but he always mentioned them to make sure that we also understand all the basic things.

[TheArtOfBeingANerd]

I love how the radical radical 5 at around 9:45 looks like it's just morphing

[michaelbaum]

You are really the best teacher I have ever had. Your videos are so clear. I live them.

[violintegral]

AMAZING integral. We get the trifecta of famous mathematical constants: e, pi and phi all in relation with one another through this integral. I found another great solution using the series expansion of cosine, the Gamma and Beta function, and the geometric series but I don't want to step on your toes by sharing too many of my own solutions. It's your channel, after all.

[qetzing]

I am simply amazed. I don't have the words to express how elegant the solution and even more so your presentation is. Truly perfect

[michaelbaum]

Fascinating solution👍It‘s so much fun watching your perfect videos. Go ahead!😀

[renesperb]

It seems worthwhile to show that for arbtrary complex a with positive realpart one has ∫ exp[-a*x^2] dx from 0 to inf = 1/2*√(π/a).This includes a number of special cases, like the one discussed in this video.

[maalikserebryakov]

Its always a spiritual experience seeing your solutions.

I feel my soul being thrown out of my body and through the cosmos. I see everything for a few seconds.

[manstuckinabox]

Involving my favorite Identity?? SWEET!

[tsa_gamer]

does anyone knows that that in I(a)= the integral from 0 to infinity of e^(-a(x^2)) dx
put ax^2=t => I(a)=1/(2sqrt of a) * gamma function of (1/2)
so for a=1-2i
we get our integral= Re(sqrt[pi/1-2i])
making the denominator real we get
sqrt(pi/5) Re(sqrt(1+2i))
=sqrt(pi/5) sqrt golden ratio

[erggish]

I have no idea how you got the results from the Laplace (and inverse) operators... But the end result of pi, phi and five was great indeed :D

[chenwong]

Where do these integral come from and how are these used in real life applications?

[fonaimartin]

Subbing u = x sqrt(1 - 2j) into the integral (of which's real part we are interested in) also works, doesn't it?
(Assuming we take the value of the Gaussian integral as known)

[manstuckinabox]

Reviewing this, 2:36, just as I was about to ask LOL.

[carlosdavid]

I got an 8 on my spanish exam, can you give me some integrals i can solve to cheer me up?

[rajibdebnath]

Hello sir, can you suggest me good mathematical physics book.

[giuseppemalaguti]

Quando sei arrivato a 2 30 basta utilizzare l'integrale della gaussiana...ma immagino te ne sarai accorto

[holyshit]

Laplace transform yes it is good idea but Euler's formula can be avoided, nevertheless nice solution