New answers tagged asymptotics
8
votes
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Series with the smallest number whose square is divisible by $n$
I couldn't find a reference, but (as noted in the OEIS page) if we have $k = a b^2$ with squarefree $a$ then $a(k) = ab$, so $$\begin{align*}
\sum_{k\leq x}\frac1{a(k)}
&= \sum_{a b^2 \leq x} \...
- 1,553
4
votes
Slick proof of Stirling's Formula?
I've played around with this a bit. I have a slick lower bound, but not a slick upper bound.
We start with the $\Gamma$-integral:
$$n! = \int_{x=0}^\infty x^n e^{-x} dx = \int_{y=-n}^\infty (n+y)^n e^{...
- 149k
1
vote
History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi
The article The origin of the method of steepest descent covers the pre-Debye history, tracing the method back to a 1829 paper by Cauchy, "Mémoire sur divers points d’analyse". Contributions ...
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