An integral that is out of this world!!

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[@TheLowstef]

There's a mistake at the bottom of the left half of the final board. The second nested "e to the e to the..." only contains one "e to the..."

[@henrymarkson3758]

Well done on reaching the 300K subscriber milestone. Thoroughly deserved.

[@demenion3521]

at 7:10 i would've written that as a complex contour integral along a semi-circle and used some residue integration to finish it off

[@ekadria-bo4962]

How?

[@aadfg0]

@@ekadria-bo4962 You can't unless there's a way of adding in a isin(θ) term. If you somehow introduce this term, integrand becomes e^(e^(iθ)) e^(iθ) dθ = -i e^z dz where z = e^(iθ), so you're integrating along upper semicircle. After that it's easy to find the value. But that's a big IF.

[@davidcroft95]

The integral at 8:32 can be easily solved if you remember that cos(n*x) is a complete set of orthonormal functions and view the integral as an extention of the dot product: therefore it's always zero unless the input of the cosine functions are equal, in this case when n=1

[@emanuellandeholm5657]

So it kinda turns into a Fourier series type cosine transform, summed over the factorial of the wave number. That was not obvious to me from the get.

[@srr5v]

You were very patient for this one. Looking forward to your new videos (though most of the time for most of them I can't understand). Still enjoy your teaching videos though, thank you for the channel.

[@holyshit922]

Answer = π/4 I tried Gauss-Legendre quadratures for 20 nodes and error was quite large

[@VideoFusco]

You can rewrite cos teta with Euler's formula and solve the two integral in a very elementary way.

[@Shindashi]

Damn this was beautiful.

[@goodplacetostop2973]

11:18

[@florisv559]

pi over 4 is a somewhat disappointing result for an out of this world integral. ;)

[@user-gs6lp9ko1c]

I would have first solved it numerically, and (probably) would have recognized the decimal equivalent of pi/4. Having that, I would have a much better chance of finding the correct answer by solving analytically. 🙂

[@s1nd3rr0z3]

Spoilers!!!

[@user-vx1kd6ks3s]

Can I suggest a bit larger hand writing on the black board? It would make it more pleasant to follow without having to constantly guess, what is being written. Thank you! And thank you for sharing this incredible result!

[@whiteboar3232]

Wow!

[@martinkausoh1386]

It always looks so easy😅

[@kinoseidon]

But what's the imaginary part of the integral appearing in 3:00?

[@cameronbigley7483]

He adds a integral of i*sin(sin(theta) etc. Essentially, a copy of the integral, but with an i attached to it, allowing for the Euler's Formula step.

[@MasterSparks42]

Can someone explain how he want from (-cos(o)+isin(o)) to (e^i(pi-o)) ? 4:25

[@yoav613]

cos(pi-(o))=-cos(o),and sin(pi-(o))=sin(o).