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Tagged with polylogarithm asymptotics
9
questions
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Asymptotic expansions of sums via Cauchy's integral formula and Watson's lemma
I am looking for references that deal with the asymptotic expansions of sums of the form
$$s(n)=\sum_{k=0}^n g(n,k)$$
using the (or similar to) following method.
We have the generating function
$$f(z)=...
- 2,217
3
votes
1
answer
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Behaviour of polylogarithm at $|z|=1$
I have the sum
$$
\sum_{n=1}^\infty \dfrac{\cos (n \theta)}{n^5} = \dfrac{\text{Li}_5 (e^{i\theta}) + \text{Li}_5 (e^{-i\theta})}{2},
$$
where $0\leq\theta < 2 \pi$ is an angle and $\text{Li}_5(z)$ ...
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1
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What's a better time complexity, $O(\log^2(n))$, or $O(n)$?
(By $O(\log^2(n))$, I mean $O((\log n)^2)$ rather than $O(\log(\log(n)) )$
I know $O(\log(n))$ is better than $O(\log^2(n))$ which itself is better than $O(\log^3(n))$ etc. But how do these compare to ...
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2
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Asymptotics of an integral involving the exponential integral
Consider the integral:
$$
I(a)=\int_a^\infty e^x E_1(x)\dfrac{dx}{x},
$$
where $a>0$ and $E_1(x)$ is the exponential integral function.
I would like to better understand the behavior of $I(a)$ for $...
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Asymptotic expansion of $\operatorname{Li}_{-x}(1/2)$
I want to find the asymptotic expansion of $$\operatorname{Li}_{-x}(1/2):=\sum_{n=1}^\infty\frac{n^x}{2^n}$$as $\Re x\to+\infty$.
My Attempt
Firstly, I considered series $$f(y)=\sum_{m=1}^\infty \...
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Polylogarithms: How to prove the asympotic expression $ z \le \mathrm{Li}_{s}(z) \le z(1+2z 2^{-s}), \;z<-1, \;s \gg \log_2|z|$
For $|z| < 1, s > 0$ the polylogarithm has the power series
$$\mathrm{Li}_{s}(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots = z\left(1+ {z \over 2^s} + {...
- 18.9k
5
votes
1
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The function $\mathrm{Li}_2(x)=\int_2^x\frac{dt}{\log^2t}$, its inverse and summation
I am reading the more understandable mathematics in the section Preliminary Results of a paper in which the authors give a explanation of facts for the logarithmic integral and its inverse. In this ...
user243301
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An asymptotic behavior of $\operatorname{Li}_{-n}(a)$ for $n\to\infty$
Suppose $a,b\in(0,1)$. I'm interested in comparison of an asymptotic behavior of $\operatorname{Li}_{-n}(a)$ and $\operatorname{Li}_{-n}(b)$ for $n\to\infty$.
Such functions exhibit approximately ...
- 47.3k
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Limit with polylog
How do you show the following limit?
$$\lim_{x\to\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac12x^2=\frac{\pi^2}3$$
Where $\operatorname{Li}_n(x)$ is the polylogarithm.
This question is ...
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