I am sure most of you are familiar with what is known as Feynman integration.
For those who are not, it allows us to integrate otherwise difficult integrals. Ones we would use
contour integration on. We choose a parameter. A 'suitable dominating
function' which we can use to make the integrand friendlier.
With this one it is not immediately seen where we can use our parameter.
But, if we rewrite it as follows:
, we can do this:
Integrate w.r.t a gives us:
Now, leaving a=1, we get:
Which, by the way, gives the result of another famous one:
I know, this is Leibniz. But it is a cool way to integrate. the trick is finding the parameter.
It works pretty slick on:
How about trying it on the famous Fresnel integrals:
by choosing the proper parameter.
I got to looking at this in depth and have a question or two. How did you just add that x in the . Is.
Now how is this useful?
So we can use this parameter to rewrite our integral as
From there it is simple
there some identity I am overlooking?. How did you actually use it to rewrite the integral?. I am not seeing something.
If , then how can:
?. Just wondering.