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Results tagged with asymptotics
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user 454
Asymptotic behavior of functions, asymptotic series and related topics
6
votes
Accepted
The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$
Let's suppose $x/f(x) \to 0$. Then take logarithm. As $x \to \infty$,
$$
\log G(x) = f(x) \log\left(1+\frac{x}{f(x)}\right)
=f(x)\left(\frac{x}{f(x)}+O\left(\left(\frac{x}{f(x)}\right)^2\right)\righ …
- 41.1k
5
votes
Does f(x)~g(x) imply $f(x) \asymp g(x)$?
My version would be this: $f(x) \sim g(x) \Longrightarrow f(x) \asymp g(x)$ is TRUE with the usual definitions, which differ from what we see above. Suppose $f, g$ are positive functions on $\mathbb …
- 41.1k
2
votes
textbooks on asymptotic expansions
http://www.amazon.com/Asymptotics-Summability-Monographs-Surveys-Mathematics/dp/1420070312/
Asymptotics and Borel Summability, O. Costin …
4
votes
Inverting an asymptotic series
There is a general technique for doing this, found in expositions dealing with transseries. One example is my own...
Transseries for beginners, Real Analysis Exchange 35 (2010) 253--310
see Pro …
- 41.1k
45
votes
Accepted
"Closed-form" functions with half-exponential growth
Yes
All such compositions are transseries in the sense here:
G. A. Edgar, "Transseries for Beginners". Real Analysis Exchange 35 (2010) 253-310
No transseries (of that type) has this intermediate gr …
- 41.1k
2
votes
Functions defined as infinite products
For the Maple computation: write
$$
1 - \frac{i}{i^{2} + n} = \frac{(i - \frac{1}{2} - \frac{\sqrt{1 - 4 n}}{2}) (i - \frac{1}{2} + \frac{\sqrt{1 - 4 n}}{2})}{(i + \sqrt{-n}) (i - \sqrt{-n})}
$$
then …
- 41.1k
0
votes
Natural candidates for sub-half-exponential which limit to half-exponential function from below
Not an answer Merely remarks.
Let me use superscript $[k]$ for $k$-fold composition. $\log^{[3]} n$ means $\log\log\log n$.
As I remarked on the other question, for fixed $a$ and $n$, the value $f(k, …
- 41.1k
4
votes
Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?
Its asymptotics are found in Chapter XXII, which leads up to the proof of the prime number theorem.
Write $\psi(x) = \log U(x)$. …
- 41.1k