Feynman’s Integration Technique is Overpowered…

In this video I use maths / the internets most favourite integration technique known as Feynman’s technique or differentiation under the integral sign to evaluate a difficult integral of sins / x from zero to infinity.
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[@jmcsquared18]

I wanted to stress test Feynman's method, so I used sin(ax) instead of introducing exp(-ax). What happens is, you get the integral of cos(ax) over [0,∞) which is undefined. But if you regularize it by introducing the regulator exp(-tx), then solve that equation, you find the regularized integral is atan(a/t) which goes to 𝜋/2 for all a>0.
That regularization is basically the exact thing you did right from the beginning, just with an extra step. So, it seems introducing exp(-ax) is the "canonically correct" way to use Feynman's method here.
Still feels kinda interesting to me two different choices for where to put the Feynman parameter end up giving similar results, if you grant the use of regularizing divergent terms as a tool.

[@SampleroftheMultiverse]

Thanks for your interesting video.
Area under a curve is often equivalent to energy. Buckling of an otherwise flat field shows a very rapid growth of this area to a point. If my model applies, it may show how the universe’s energy naturally developed from the inherent behavior of fields.
Your subscribers might want to see this 1:29 minutes video showing under the right conditions, the quantization of a field is easily produced.
The ground state energy is induced via Euler’s contain column analysis. Containing the column must come in to play before over buckling, or the effect will not work. The sheet of elastic material “system”response in a quantized manor when force is applied in the perpendicular direction.
Bonding at the points of highest probabilities and maximum duration( ie peeks and troughs) of the fields “sheet” produced a stable structure when the undulations are bonded to a flat sheet that is placed above and below the core material.
Some say this model is no different than plucking guitar strings. You can not make structures with vibrating guitar strings or harmonic oscillators.
ruclips.net/video/wrBsqiE0vG4/видео.htmlsi=waT8lY2iX-wJdjO3
At this time in my research, I have been trying to describe the “U” shape formed that is produced before phase change.
In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level.
Over-lapping all frequencies together using Fournier Transforms, can produce a “U” shape or square wave form.
Wondering if Feynman Path Integrals for all possible wave functions could be applicable here too?
If this model has merit, seeing the sawtooth load verse deflection graph produced could give some real insight in what happened during the quantum jumps between energy levels.
The mechanical description and white paper that goes with the video can be found on my LinkedIn and RUclips pages.
You can reproduce my results using a sheet of Mylar* ( the clear plastic found in some school essay folders.
Seeing it first hand is worth the effort!

[@DarkBidhan]

Wow. That’s a great video.

[@abhijit5290]

Wonderfully explained. Loved It!

[@shaiyanraquibchowdhury7466]

Loved the explanation

[@shreebhattacharjee3502]

Love the bow