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Results tagged with asymptotics
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user 37436
Asymptotic behavior of functions, asymptotic series and related topics
8
votes
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
Edit: I corrected a typo in the coefficient of $x^{-8}$ in the expansion of $n_{0}$ for the asymptotics of $C(x)$. …
- 1,273
8
votes
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
.
$$
Now exchange the summation of the asymptotics in $n$ (with, say, summation index $k$) and the summation over $n$, which I assume light heartedly to be possible. …
- 1,273
6
votes
Accepted
Exponential approximation for 3F2 hypergeometric function with repeated indices
Indeed the hypergeometric series is unimodal (the terms for fixed $z$ and $d$ first grow with $k$ then decrease again). The hypergeometric series is given by
$$
{}_{3}F_{2}(d,d,d;d+1,d+1;z)=\sum_{k=0} …
- 1,273
5
votes
Accepted
expression for infinite series with powers of factorial in denominator
I copied this from my comments above:
This sum was discussed in the book "Advanced Mathematical Methods for Scientists and Engineers" by Carl M. Bender and Steven A Orszag in section 6.7 Example 4. T …
- 1,273
3
votes
Accepted
Asympotics of $\int_0^{\infty} \left[ Q(m,x)\right]^d dx$
Short:
The answer for $d=3$ is $a=\frac{3}{2\sqrt{\pi}}=0.846283 \dots$.
Long:
The integral is splitted in two at $x = m$.
$$
\int_{0}^{m} \ Q(m,x)^d \ dx + \int_{m}^{\infty} \ Q(m,x)^d \ dx
$$
For …
- 1,273
2
votes
Limit of a hypergeometric integral
Not a full proof, but a "closed" formula with two independent finite sums. I am sure that the proof can be made more compact, but I hadn't have the time to think about it.
Very important: I assume wi …
- 1,273
2
votes
Asymptotic behavior of integral with gamma functions
Too long for a comment:
By using the approach of @Carlo Beenakker I was able to get an asymptotic formula for $z=\frac{1}{2} + z_{0} e^{i \phi}$ with $z_{0}\rightarrow\infty$ and $-\frac{\pi}{2}<\phi …
- 1,273
2
votes
Accepted
Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $...
You have done much of the work for yourself. Here is the last missing step:
Writing the $_2 F_3$ as integral (see, e.g., http://dlmf.nist.gov/16.5.E1):
$$
_2 F_3(x+1,x+1;1,1,1;\alpha)=\Gamma(x+1)^{-2 …
- 1,273
1
vote
Asymptotic value of the Shannon entropy
Laplace's Method: For large $N$ the summand as function of $n$ is maximal at $n=\frac{N}{2}$. This can be seen by the Laplace-deMoivre approximation of the binomial
$$
{N \choose n}\sim 2^{N}\sqrt{\fr …
- 1,273
1
vote
Asymptotic growth of recurrence relation $x_n=\min\limits_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_...
For $a>2$ it has the more complicated asymptotics, $O(n a^{\ln n/\ln 2})$ (Edit: or, what is the same, $O(n^{1+\ln a/ \ln 2})$). …
- 1,273
0
votes
Accepted
Estimate the scale of the power series with Poisson pdf/pmf-like terms
Actually this can be solved by standard asymtotic analysis methods (with help from Mathematica):
Expand the factorial in the sum with Sterling. Then do as if $k$ is continuous and find $k=k_{0}$ wher …
- 1,273