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
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
