Tagged Questions
2
votes
0answers
33 views
Limit involving incomplete gamma function
Let $\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} dt$ be the incomplete gamma function. What is the limit
$$
\lim_{p \to \infty} \frac{1}{\sqrt{p}} \Big[\Gamma(p+\tfrac12,x)\Big]^{\frac{1}{2p}}
$$
as a ...
2
votes
1answer
99 views
Limit involving a hypergeometric function
I am new to hypergeometric function and am interested in evaluating the following limit:
$$L(m,n,r)=\lim_{x\rightarrow 0^+} x^m\times {}_2F_1\left(-m,-n,-(m+n);1-\frac{r}{x}\right)$$
where $n$ and ...
2
votes
1answer
65 views
Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$?
I'm trying to give at least some partial answers for one of my own questions (this one).
There the following arose:
$\hskip1.7in$ Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm ...
5
votes
1answer
129 views
Error Function limit
$$\prod_{n=1}^{\infty}{\frac{2}{\sqrt{\pi}}\int_0^n e^{-x^{2}} \mathrm{d}x} \approx 0.83874 $$
Is it a known constant? I couldn't find anything about it. Do you know ways to calculate the value ...
0
votes
1answer
51 views
How to find $\lim_{x \rightarrow -N} J_{a} = (-1)^N J_{N}$?
So far I have $$\lim_{a \rightarrow -N} J_{a} = \lim_{a \rightarrow -N} \mid \frac{x}{2}
\mid^a \sum_{k=0}^{\infty} \frac{(-x^2/4)^k}{k! \; \Gamma(a+k+1)} = \mid \frac{x}{2}
\mid^{-N} ...
0
votes
1answer
152 views
Limit of a summation wth Gamma function
Can anyone prove this (I'm very confident that it is correct) or have any idea how this can be handled:
$$
\lim_{n \rightarrow \infty} \frac{1}{n-1}\sum_{i=1}^{n-1} \frac{1}{(\alpha-1)(n-i) -1} ...
1
vote
1answer
59 views
Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$
Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$
Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W
...
7
votes
2answers
194 views
How to prove asymptotic limit of an incomplete Gamma function
How can I prove:
$$\lim_{z\to\infty}{\Gamma(z, x)\over\Gamma(z)} = 1$$
? Here $\Gamma(z, x)$ is the upper incomplete gamma function and $\Gamma(z)$ is the gamma function.
This must be something ...
0
votes
1answer
65 views
Limit of an integral containing a product
I'm stuck on the following problem:
given the integral:
$$I(N)=\int \prod_{k=1}^N \left(k-\frac{k}{x}\right) \, dx$$
calculate the following limit:
$$I_{\infty}(N)=\lim_{x\to\infty}I(N)$$
I know that
...
4
votes
1answer
72 views
Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$
How to show that
$$
\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1
$$
where $\psi(x)$ is the digamma function.
2
votes
2answers
111 views
Compute $\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$
What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I ...
16
votes
3answers
357 views
An interesting sum to infinity
Is there any simple way of computing the following sum?
$$\sum_{k=1}^\infty \frac1{k\space k!}$$
2
votes
1answer
146 views
Limit of the function $\zeta(x)/\zeta(x+1)$ as $x \to \infty$
I am looking for a simple proof that $\zeta(\alpha)/\zeta(\alpha+1) \to 1$ as $\alpha \to \infty$ (where $\zeta(\alpha)$ denotes the Riemann zeta function, $\zeta(\alpha) = \sum \limits_{n\geq 1} ...
5
votes
1answer
609 views
Connection between Legendre polynomial and Bessel function
In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation
$$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$
valid for $x>0$.
...
2
votes
1answer
183 views
Intuitive limit. Integration and $\gamma(n,x)$
It isn't hard to prove that:
$$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$
Or put in a different way:
$$\int_0^x e^{-t} \frac{t^n}{n!} dt = ...
5
votes
3answers
282 views
Limit of Zeta function
I'm looking for a reference for (or an elementary proof of)
$$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$
Thanks for your help.
1
vote
1answer
145 views
Closed form formula for series involving derivatives of reciprocal gamma function
How to get closed form for the sum $\displaystyle{\sum\limits_{k = 1}^\infty {\frac{{{p^k}}}
{{\left( {2k} \right)!!}}\frac{{{d^k}}}
{{d{s^k}}}{{\left. {\frac{1}
{{\Gamma \left( s \right)}}} ...
1
vote
1answer
97 views
Find function if known some limit
I have trouble with the following problem.
Let $f=f(p)$, $p>1$, $0<f<1$, and $\lim_{p\to\infty}f(p)=1$.
Find $f(p)$, such that
$$\lim_{p\to\infty}p\left(1-\sqrt{p ...
5
votes
1answer
136 views
Lower bounding a ratio of gamma functions
I am trying to show that the following function has a lower bound of $\ \frac{1}{2}$ for all $c\geq 2$. Or, alternatively, that that function increases with $\ c$:
...
4
votes
2answers
97 views
Limits of a function involving $\mathrm{cn}(x,k)$
Given $$f(x) = \frac{1 - \mathrm{cn}(x,k)}{{\sqrt3}(1+\mathrm{cn}(x,k)) - 1 + \mathrm{cn}(x,k)}$$
what would be $$\lim_{x\to 0} f(x)$$
and $$\lim_{x\to\infty} f(x)$$ when
...
3
votes
1answer
120 views
Asymptotic approximation for confluent hypergeometric function
I have the following nasty expression that I would like to expand in powers of $\frac{1}{N}$:
\begin{align}
\frac{2^{\frac{3}{2}} 3^{\frac{1}{2}} \Biggl[ \sqrt{u} \cdot ...
