An Insane Approach! The Fresnel integrals: sin(x^2) and cos(x^2) without complex analysis

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In today's video we are going to play around with the integral of e^(-ix^2) from -infinity to infinity. With this one integral we are going to be able to evaluate the two Fresnel integrals. But not in the complex way! Let's solve it by using algebraic methods and differentiation under the integral sign, also known as the Leibniz rule for integrals or Feynman integration. Have fun =)

[Ricocossa]

17:13 I disagree, it would be true if the exponent was real, but in this case it is imaginary, so the exponential retains a modulus of one. You have to do more work to prove that the integral goes to 0.
My guess is that there's something to do with distributions, or Fourier transforms.

[martinepstein]

Wow! This was amazing. There's one step that seems to be missing. You have two options for sqrt(-i) that are negatives of each other. The way I justify your choice is by rewriting the sin(x^2) integral as twice an integral from 0 to oo, then noticing it must be positive by viewing it as an alternating series of the areas under each hump.

Come to think of it, of course the integrals for cos(x^2) and sin(x^2) are equal since sin(x^2) = cos(x^2 - pi/2) and a translation by pi/2 doesn't matter when the bounds are infinite.

[stevethecatcouch]

Your English is quite good. I once had a professor with so thick a Polish accent that I couldn't understand half of what he said. Fortunately, it was a math class so I was able to follow most of what he said based on what he wrote on the board. Great videos.

[FernandoVinny]

before this "c" I really thought that the constant wasn't important

[palahnuk]

I love this guy - I suck at filming videos and speaking English. Very funny and humble. Nicely presented. You took the time to clearly step by step explain the approach here.

[Arycke]

Your videos are AWESOME. I have a degree in mathematics and I've been on the alternative side of the internet and life for over a decade. Therefore, your usage of profanity, dank usage of memes (no normie shit), and some tough integral bois compel me to watch your videos. Thank you for existing :o The substance, thoroughness, and demeanor make these videos so fun to watch and enjoy!

[MegaTRIANGULUM]

This are the kind of math channels we were looking for!! Keep up the great work! :D

[enisbarkngoktas]

This is an excellent channel prepared with really high amount of effort clearly. Thank you so much for providing this amazing platform for us!

[erfanmohagheghian]

FUCK!!! limit of exp(-i*x^2) x-> inf is not zero, it's undefined. cos(x^2) -i*sin(x^2) does not converge at inf!
You were just lucky that you got the correct solution. Fix it babe. I honestly did not expect you to make such a mistake.

[edwinlin]

Great video as always. :D
You mention at the end that you were working on this for a long time. How exactly did you go about figuring out a way to solve for these. Like how did you get to your initial I(t)? Specifically how to recognize that putting the t in the bounds is better than, say, multiplying some function in terms of t to the integrand.

Like I know a common way for the dirichlet integral (sin(t)/t) is to multiply in an e^(-st) and differentiate that under the integral sign, and that one is kinda justified from the laplace of sin(t)/t, but for this video it just seems so Out of Nowhere.

This is really a genius way to approach it.

[FineFlu]

Your English is great, your video quality is great, your mathematics is great.

I enjoyed this a lot, especially because I could follow you, for the most part.

[pacolibre]

Not to party poop, but cant you solve the integral of e^(-ax^2) with the polar coordinate transformation for the Gaussian integral, get sqrt(pi/a) and just plug in a=i?

[hachkoko]

At 4:00 you can just use the fundamental theorem of calculus, which tells you that what's inside the square is an antiderivative of t->exp(-it^2)

[carlosgiovanardi]

20:00 - > For sure I enjoyed a lot. Great work!

[BrainsOverGains]

At 17:35 isn't √ab =√a√b only valid for positive a and b?

[thomasq]

I attempted this integral on my own and I was able to do it in less than 5 mins. You just need to change exp[-i(x)^2] to exp[-(√i*x)^2] and let u=√i*x. After that you get a constant multiplied by the gaussian integral. Then just calculate i^(-0.5) and you get the final result.

[Afdch]

14:30 Sorry for being dumb and stuff but what's up with making t=0 AND having u = x/t at the same time? Why is this legit for finding C?

[HoldensBro]

You could also just use the standard result of the Gaussian integral and make a u sub for u=xsqrt(i) and you’re basically there

[Davidamp]

Let u= xsqrti, du= sqrti dx. Integral becomes: 1/sqrti integral from -inf to +inf of e^-(u^2). That's all.

17:20 Uh exp(-i*inf) is not zero. The complex exponential oscillates and never changes amplitude. Just like cos(x) doesn't tend to zero as x tends to infinity. If your integral was on the imaginary line and not the real line then I think you could say it goes to zero, but why are you saying that an oscillatory integrand tends to zero?