I wanted to remark that the "fake function" idea is generalizable.
We can find any entire asymptotic with nonnegative derivatives by simply using a rising integer function T(n) :
\( f(x) = \sum_{n = 1}^{\infty} a_{T(n)} x^{T(n)} \; < \; \exp^{0.5}(x) \)
\( \forall {x \gt 0 } \; \; a_{T(n)} x^{T(n)} \; < \; \exp^{0.5}(x) \)
\( \forall {x \gt 0 } \; \; \log(a_{T(n)} x^{T(n)}) \; < \; \log(\exp^{0.5}(x)) \)
\( \forall {x \gt 0 } \; \; \log(a_{T(n)}) + {T(n)}\log(x) \; < \; \log(\exp^{0.5}(x)) \)
etc.
This simple idea / equation is very powerfull.
The fundamental theorem of fake function theory.
In combination with the post about the inverse gamma function we could for instance find the asymptotic :
\( f(x) = \sum_{n = 1}^{\infty} \frac{x^{T(n)}} {T(n) !} \)
with the method above.
To stay in the spirit of the exp.
regards
tommy1729
We can find any entire asymptotic with nonnegative derivatives by simply using a rising integer function T(n) :
\( f(x) = \sum_{n = 1}^{\infty} a_{T(n)} x^{T(n)} \; < \; \exp^{0.5}(x) \)
\( \forall {x \gt 0 } \; \; a_{T(n)} x^{T(n)} \; < \; \exp^{0.5}(x) \)
\( \forall {x \gt 0 } \; \; \log(a_{T(n)} x^{T(n)}) \; < \; \log(\exp^{0.5}(x)) \)
\( \forall {x \gt 0 } \; \; \log(a_{T(n)}) + {T(n)}\log(x) \; < \; \log(\exp^{0.5}(x)) \)
etc.
This simple idea / equation is very powerfull.
The fundamental theorem of fake function theory.
In combination with the post about the inverse gamma function we could for instance find the asymptotic :
\( f(x) = \sum_{n = 1}^{\infty} \frac{x^{T(n)}} {T(n) !} \)
with the method above.
To stay in the spirit of the exp.
regards
tommy1729
