This integral will improve your advanced math skills

A beautiful integral solved using Feynman's technique and complex numbers. Full solution development leading to a result involving pi and the euler mascheroni constant.

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Advanced MathWear:

[_DD_]

Love your content. Really good stuff!! No time wasted, straight to the point! Great channel!

[MikeB-qv]

Hi, you might consider that sin(x)/x = integral(0, 1)cos(a*x)da this removes the issue with the constant of integration and does not require taking the imaginary part of a logarithmically divergent expression. Of course you got the right answer!

[MrWael]

Thank you for your stunning solution. It is very interesting.

[wohargRadu]

This would be nice if you can switch derivation and integration but for that the integral must converge.
As Exp(-x).sin(x) . ln(x) /x ~1.1. ln(x) = ln(x) when x->0 it would have been correct to say that the integral of ln (x) converges in the neighbourhood of 0 what justifies the switch because the integral converges trivially in + infinity.

[SuperSilver]

I’ve been encountering this problem. One where Feynman’s trick is resulting in crazy divergence. One thing I think that remedies this problem is integrating the expression with respect to your integration parameter while you are in the middle of the setup of your anti derivative. In this case here’s what I mean (I’m gonna use v for the E-M constant)

I’(a) = (v+ln(a))/(a)

1/a = int(exp(-ax)) from 0 to inf

Let’s use that here

I’(a) = int((v+ln(a))*exp(-a*x)) from 0 to inf

Now integrate with respect to a, and interchange the order of integration

I(a) = from 0 to inf + C

Here I have performed the integration over a, and invoked a special function. I think we like to avoid these, but the nature of the problem maybe dictates its purpose.

Now just take the limit as a goes inf, C = 0 pops out rather trivially, as I(inf) is also equivalently zero. Then we can go back to our derivative and proceed as usual with the Laplace transform definition we had.

This is at least a framework for how I would consider the problem of solving for the constant. I didn’t really prove you could interchange the order of the integration like that and then take the limits under the integral sign, but it does work so maybe that’s something.

[cameronspalding]

What you have defined as I(alpha), I would have called J(alpha)

[energeticgorilla]

what software/app do you use for these videos? its very pretty and i would love to use it myself 😊

[seesky]

sinx=?*e^ix, how does it become like this?

[shivamdahake]

Ignoring the C, you are tresspassing into the world of physicists my

I suggest you turn back, its for your own safety.