A RIDICULOUSLY AWESOME INTEGRAL!!! real vs complex methods with @qncubed3

This video answers a question as old as time itself; is there an integral that has an absolutely gorgeous solution using both real and complex methods of integration?

Well here's that integral along with 2 wonderful solution developments with real and complex methods by me and qncubed3 respectively.

[slavinojunepri]

Two great solutions from two math grand masters. Thank you so much!

[Observer_detector]

Complex analysis Technic Integral = Easy way 🤗🤗

[violintegral]

Beautiful presentation of beautiful solutions of beautiful integrals. Doesn't get much better than this!

[manstuckinabox]

Let's GOOO!!! I missed you ;.; glad you're back with some cool company :D

I don't think (16:27) that using complex formulas to solve real integrals is considered part of complex analysis, we're just using a formula that is also defined properly within the real axis, since the parameter t is quite real, and yeah, I do agree that the Euler formula stands firmly between Real methods and the complex methods.

I'm defenitly archiving the real method, such a nice neat way to takle an integral

[violintegral]

I was able to show that this integral is equal to the integral from 0 to pi/2 of w.r.t. x using the series expansions of sinh(x) and cosh(x), Lobachevsky's integral formula, and a related analogous formula for the integral from 0 to infinity of sin(x)/x*f(x) w.r.t. x. Both Wolfram Alpha and Desmos seem to verify my result by numerical integration, however I struggled to proceed with the integration beyond this point. I tried to make use of Feynman's trick but I had no luck. Perhaps Contour Integration will work from this point on, but I'm still wondering if there is a solution using purely real methods. Anyone have any ideas?

[zunaidparker]

This was very cool. Love the presentation of two different methods! Points to both techniques, but the win to qncubed3 for presentation. Very clear explanation and easy to follow the structure of the pieces.
Maths505 was on a good track but then got lost in a tangent explaining the mistake you made instead of completing the proof straightforwardly by building the k=0 case into it in a natural way.

[rajendramisir]

The semi-circle contour approach to the solution was well thought out, executed and explained. The real analysis using Euler’s Formula approach was swift and clever for this type of integral. Brilliant logical reasoning, execution and explanation by Math505. If our existence depended on a quick solution to this integral then Math 505 would have secured our existence. Smart complex & real men!

[rajendramisir]

I am impressed by both solutions to this integral. I think, with such developed and analytical minds, Math 505 and qncubed3 should ponder The Reimann’s Hypothesis and dive into the nano universe of Quantum Field Theory & Quantum Mechanics. I suspect you guys are PhDs.

[renesperb]

Both techniques are very useful and one can learn a lot from them. The result is also very nice. By the way : Mathematica, which is in general very good in
calculating integrals, can't do this one.

[hectore.garcia]

I think that in the second part you made illegal tricks like commuting the integral and the summation(sigma), this can be done for all k where the function is uniform. For the exponential function it is uniform but it would have to be proved.