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Searching for an asymptotic to exp[0.5]
Forgive me for not using tex again.

On my to do list are considering different solution to

fake(x^x)

fake(x^ln(x))

fake(exp(x) ln(x)^2)

fake(x^sqrt(x))

But for now I was mainly intrested in :

fake(Gamma(x+2))

In particular we can use for instance the the fake log or fake sqrt results here !

fake Gamma(x+2) = integral_0^oo fake( exp(t) t^(x+1) ) exp(-2t) dt

where fake ( exp(t) t^(x+1) ) = Mittag(t,1,x+1) as obtained before.

( fake exp(x)sqrt(x) = Mittag(x,1,1/2) as example )

Notice the almost self-similarity , Mittag depends on the gamma function !
( A tempting idea is to replace the gamma in the Mittag function with fake gamma itself ?! )

Remember that there is analytic continuation for the integral !

Also if we use the Cauchy integral on
integral_0^oo fake( exp(t) t^(x+1) ) exp(-2t) dt

Then we have some fine looking calculus expression imho.
Many tricks from standard calculus could probably be used such as interchanging the integrals , or (interchanging) a sum and an integral , or feynman integration etc. and of course contour integration techniques/theorems.

Combining integral transforms with fake function theory seems intresting both for computation and theory.

Many complicated functions can be given by contour integration of simpler ones , and those contours can be written as simpler integral transforms.
Those transforms can then contain a sqrt or ln or such and therefore we finally arrive at a fake function of the Original complicated one.
- For instance g(z) , the functional inverse of a function f(z) such that g(z) grows fast enough ( faster then poly ) and g(z) cannot be given by elementary functions -

Probably fake( Gamma(x+2) ) has a closed form or its derivatives at 0 have a closed form.

There has already been research done into asymptotics to the gamma function but not like this.

Probably the Gamma function will popularize fake function theory.

Generalizing recursion and/or functional equations into fake function theory might be the next step.

The number of roads this leads too is uncountable !


Next on the list is fake( exp(x) x / ln^2(x) ) exp(-x) without using a fake ln.

regards

tommy1729
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Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/18/2014, 11:07 PM

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