All Questions
Tagged with gamma-function asymptotics
126 questions
5
votes
0
answers
191
views
An asymptotic for a sum involving gamma functions
I have a rather scary sum
$$S(n) = \sum _{k=0} ^{\infty} \frac{\Gamma^A\left( k+an+1 \right)\Gamma^B\left( k+bn+\frac12 \right)}{\Gamma^{A-1}\left( k+1 \right)\Gamma^{B-1}\left( k+\frac12 \right)}\...
- 597
4
votes
1
answer
130
views
Large $x$ behavior for the sum $\sum _{m=1} ^{\infty} \frac{x^{m(A+B)}}{\Gamma(m+1)^A\cdot \Gamma \left( m+\frac12 \right)^B}$
Consider the sum $S$ given by
$$S_{A,B}(x) =\sum _{m=1} ^{\infty} \frac{x^{m(A+B)}}{\Gamma(m+1)^A\cdot \Gamma \left( m+\frac12 \right)^B}, \quad \text{where}~~ A,B \in \mathbb{N}$$
Wolfram Mathematica ...
- 597
0
votes
2
answers
56
views
Sitrling's formula for gamma function with negative argument
We have the classical Stirling's approximation formula for the $\Gamma(\cdot)$ function in the form:
$$
\Gamma(x)\approx\sqrt{2\pi}\exp{(-x)}x^{x-1/2}
$$
for $x >0 $ as $x\to\infty$.
I am ...
- 365
0
votes
1
answer
54
views
Asymptotic of incomplete Beta function
Let $m,n \in \mathbb{N}$ with $m \le \log n$.
I want to find the sharp range of $d_{m,n}$ such that there exists constant $c$ $\textbf{independent}$ $\textbf{of}$ $m,n$ and
$$
\frac{1}{c}\,\mbox{B}(m,...
- 133
2
votes
0
answers
57
views
Upper bound for Gamma function in Tenenbaum's book
I am recently reading Tenenbaum's monograph Introduction to Analytic and Probabilistic Number Theory. When I am studying the Chapter of Euler Gamma function, I encounter this exercise:
Show that for ...
- 41
0
votes
2
answers
111
views
Asymptotic expansion / behaviour of integral function at large x [closed]
How can I find the asymptotic expansion of the following integral, i.e. its behavior for large $x$?
$$ \int_0^x \frac{y^k}{\sqrt{1+y^l}}dy $$
I know that the integral can be solved exactly for certain ...
- 11
0
votes
1
answer
99
views
Proof using the Stirling formula for a limit behaviour
I am trying to understand/prove a lemma which is stated in two papers about branching processes (Lemma 2.2 in "Martingales And Large Deviations For Binary Search Trees" by Jabbour-Hattab and ...
- 308
1
vote
2
answers
104
views
Asymptotic expansion using gamma functions and Stirling's formula
Question: obtain an expression for the $n$th term of an asymptotic expansion, valid as $ \lambda \to \infty$ for the integral
$$I(\lambda) = \int_0^1 t^{2\alpha}e^{-\lambda(t^2+t^3)} dt $$
where $ \...
- 305
0
votes
0
answers
35
views
Showing the asymptotic behavior of the expectation of a function of a Galton-Watson process using the Stirling formula
So right now I am trying to understand the proof in this paper. Let $(Z_n)_{n\in\mathbb{N}_0}$ be a Galton-Watson process with infinite offspring mean (I think supercriticality works as well), $\xi>...
- 78
0
votes
2
answers
46
views
Limit of sum of ratios of gamma function like products
I'm interested in evaluating:
$$\lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{1}{i^2} [ \prod_{j=i}^n \frac{kj -1}{kj} ]^2$$
for $k$ some constant. After trying a bit, I have not figured out if the ...
- 95
2
votes
1
answer
100
views
Stirling's Formula for Quotient of Gamma Functions
I am reading a paper in which the authors arrive at the quotient of Gamma functions $\frac{\Gamma(s+k-1)}{\Gamma(k-1)}$ where we are thinking of $s$ as the variable and $k$ is just some other ...
- 2,340
1
vote
0
answers
61
views
Asymptotic expression of the maximum of the relative error of the Fourier expansion of the multifactorial
I wanted to study the following problem:
Let
$$x!_{(a)}=a^{\frac{x}{a}}\Gamma\left(1+\frac{x}{a}\right)\prod_{j=1}^{a-1}\left(\frac{a^{\frac{a-j}{a}}}{\Gamma\left(\frac{j}{a}\right)}\right)^{C_{a}\...
0
votes
0
answers
155
views
Asymptotic behavior of Gamma cdf, gamma function, and incomplete gamma function
For the sake of this post, we only look at real-valued gamma functions.
We know that $\lim_{b \to \infty}\gamma(a,b)=\Gamma(a)$, so we have
$$\lim_{b \to \infty} \frac{\gamma(a,b)}{\Gamma(a)}=1.$$
How ...
- 107
5
votes
1
answer
176
views
What's the intuition behind $\frac{\sqrt{\pi}\left(\frac{x}{e}\right)^{x}\left(8x^{3}+4x^{2}+x+\frac{1}{30}\right)^{\frac{1}{6}}}{x!}>1,\forall x>0$?
As the title say I try to better understand the approximation due to Ramanujan :
$$\frac{\sqrt{\pi}\left(\frac{x}{e}\right)^{x}\left(8x^{3}+4x^{2}+x+\frac{1}{30}\right)^{\frac{1}{6}}}{x!}>1,\forall ...
- 3,742
1
vote
0
answers
109
views
Big O subscript notation
I am reading a paper in which a proof say that for $n \geq 1$ and $|z| \leq A$ (where $z \in \mathbb{C}$) the following function is bounded as follows:
$$ \frac{\Gamma(n+z)}{\Gamma(n+1)}= \frac{n^z}{n+...
4
votes
2
answers
542
views
Solving for $x$, $e^x=x!$
I am curious on where the factorial $x!=\Gamma(x+1)$ passes the exponential ${\rm e}^x$ for $x>0$. I first tried using Lambert $W$ and the inverse of Stirling's approximation by @Gary. Call $p$ the ...
- 2,479
0
votes
1
answer
133
views
Showing $\Gamma'(1)=-\gamma$ using recursion property of $\Gamma(x)$
From the book "Asymptotic Analysis and Perturbation Theory" by William Paulsen,
Problem 23 p.67.
Show that $\Gamma'(1)=-\gamma$.
Hint: First use the recursion property of $\Gamma(x)$ to ...
- 2,479
1
vote
1
answer
103
views
Asymptotics of incomplete gamma function
Let $\Gamma(a,x)$ denote the incomplete Gamma function. Consider the function
$$
f(x)=x^{-x}\Gamma(x,x)-\Gamma(0,x)=x^{-x}\Gamma(x,x)-E_1(x),
$$
where $E_1(x)$ denotes the exponential integral. I ...
- 775
3
votes
1
answer
155
views
Asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$?
What are asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$ ?
Here is the graph for $n=10$.
The turning point in $x\in(0,1)$...
- 30k
6
votes
1
answer
186
views
$\lim_{n\to\infty} \left(\frac{e}{n}\right)^n \int_0^n |x(x-1)(x-2)\dots(x-n)|dx$
What is
$$
\lim_{n\to\infty} \left(\frac{e}{n}\right)^n
\int_0^n \left| x(x-1)(x-2) \cdots (x-n) \right| \, dx?
$$
Context: I was trying to find an asymptotic expression for the total area of the ...
- 30k
2
votes
2
answers
362
views
How to Integrate the Gamma Function
Problem: Evaluate $$\int \Gamma(x)\, \mathrm{d}x$$
In asymptotic analysis, functions are compared with each other in terms of growth.
I want to know how much better one function is than another.
To ...
0
votes
1
answer
214
views
Asymptotic Approximations for Higher-Order Factorials (e.g. triple factorial) and the Gamma Function
I recently found the asymptotic expansion for $\frac{(2n)!!}{(2n-1)!!}$ to be $\sqrt{\pi n}$ by simplifying the double factorials and applying Stirling's formula.
However, I was unable to find an ...
- 537
1
vote
1
answer
177
views
Improving the Stirling's formula .
Do we have :
$$\frac{x!}{\sqrt{2\pi x}\left(\frac{x}{e}\right)^{x}e^{\frac{1}{12x}}}\left(1+\frac{1}{6}\ln\left(8x^{3}+4x^{2}+x+\frac{1}{30}\right)-\ln\left(\sqrt{2x}e^{\frac{1}{12x-\frac{1}{\pi^{2}x^{...
- 3,742
6
votes
3
answers
189
views
Asymptotic behaviour of Gamma functions
I would like to simplify or get a simpler approximation of the following ratio involving the Gamma function
\begin{equation}\tilde{\gamma}(K)\triangleq \frac{1}{\sqrt{K}}\frac{\Gamma\left(\frac{3K}{2}...
- 753
0
votes
1
answer
81
views
Prove $\frac{\Gamma(t+a)}{\Gamma(t)} = t^a(1+O(1/t)$ using Stirling's formula
Let $t$ be positive integers and $a$ is fixed. Stirlings' formula for the Gamma function yields
$$
e^{-t}t^{t+\frac{1}{2}}\sqrt{2\pi} \leq \Gamma(t+1) \leq e^{-t}t^{t+\frac{1}{2}}\sqrt{2\pi}e^{\frac{1}...
- 961
1
vote
2
answers
91
views
How can I calculate a limit involving the Gamma function?
I need to find the following limit:
$$
\lim_{n \to \infty} \frac{\Gamma\left(\frac n2\right)}{\Gamma\left(\frac{n+1}2\right)}.
$$
Wolfram Alpha tells me it is 0. Also Wikipedia told me that for $x$ ...
- 587
-1
votes
2
answers
124
views
New formula like Stirling formula for product of Gamma function :$ \left(x!\right)\left(\frac{1}{x}\right)!$? [closed]
Inspired by a problem of my book wich is :
Let $0<x$ then we have :
$$f(x)=\left(x!\right)\left(\frac{1}{x}\right)!\geq 1$$
I have tried several function (log,LambertW,$x^a/x+1$,...) but now I ...
- 3,742
3
votes
2
answers
391
views
Asymptotic expansion of the gamma function with imaginary value
For fixed $h>0$, consider the function
$$F(x) = e^{-x\pi/2} \left| \Gamma\left(h-i\frac x2\right)\right|^2.$$
What is its asymptotic expansion as $x\to 0$ or $x\to \pm\infty$?
For $x\to \pm\infty$, ...
- 2,206
3
votes
0
answers
302
views
Saddle point approximation
The coefficients for the Maclaurin series of the reciprocal $\Gamma$ function can be calculated by an integral formula given by:
$$\frac{1}{\Gamma(z)}= \Sigma_{n} a_n z^n, \;a_n =\frac{(-1)^n}{\pi n!} ...
- 107
2
votes
3
answers
97
views
Ratio of $\Gamma(n)$ to $\Gamma(n/2)^2, n\to \infty?$
I'm trying to find the limit
$$\lim_{n\to \infty}\dfrac{\Gamma(n)}{\Gamma(\frac{n}{2})^2},$$
and also trying to find the precise decay rate for the above, i.e. the function $D(n)$ so that if $G(n):=\...
- 2,556
2
votes
2
answers
104
views
Equivalence of Stirling's approximations for $n!$ and $\Gamma(z)$
In this Wikipedia article for Stirling's approximation,
it is first shown that
$$ n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1+O\left(\frac{1}{n}\right)\right)$$
for $n\in \mathbb N$,
and then it ...
- 378
2
votes
1
answer
95
views
Asymptotic series for $\ln\binom{x}{k}$ as $k\to\infty$
I was playing around with $\binom{x}{k}$ for fixed $x$ as $k\to\infty$ and realized it could be compared with the Weierstrass factorization of the reciprocal $1/\Gamma(x)$ to get $\binom{x}{k}\sim k^{-...
- 154k
2
votes
0
answers
68
views
Can You Perform Operations on a Big O Term
I am a student and have been reading about Stirling's approximation of the gamma function. Wikipedia has a result as follows
$$\Gamma(z) = \sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z\left(1 + O\...
- 51
-1
votes
1
answer
122
views
$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$
I want to prove this formula using Stirling's approximation or otherwise:$\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ wher $s=\sigma+it$.
Can someone please ...
user1073119
2
votes
1
answer
79
views
Computing the asymptotics of a principal value integral
I have been looking at the following principal value integral (with $x>0$ and $0< \sigma < 1$)
$$
\mathrm{P.V.} \int_0^\infty \frac{v^{-\sigma} e^{-x v}}{1 - v} dv,
$$
and I would like to ...
- 185
0
votes
0
answers
204
views
Asymptotic behavior of modified Bessel function help
The modified Bessel function of the first kind is defined as
$$I_q(\rho)=\sum_{m=0}^{\infty} \frac{\left(\frac{\rho}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}$$ where $\rho \in \mathbb{C}\setminus \{0\}$ and ...
- 323
3
votes
1
answer
124
views
"Tipping point" between asymptotic behavior of gamma function along the line $z=x+mxi$
Disclaimer: I am an undergraduate student about a semester into introductory complex analysis. I am entirely out of my depth here, just curious about something I noticed.
The gamma function $\Gamma(z) ...
- 459
2
votes
0
answers
138
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions ( class of functions ) . In doing so this just pops out and couldn't handle the said integral so asked ...
- 191
2
votes
1
answer
62
views
Continuation of an asymptotic series originally defined for $z>0$ to $z<0$
Consider the well-known asymptotic expansion of the $\Gamma%$-function:
$$\Gamma(x)\sim\left(\frac{x}{e}\right)^x \sqrt{\frac{2 \pi }{x}}\left[1+\frac{1}{12 x}+\frac{1}{288 x^2}-\frac{139}{51840 x^3}-\...
- 47.6k
9
votes
1
answer
332
views
How might one make this Gaussian integral derivation of Stirling's approximation more airtight?
I will be following an argument presented by the YouTuber Flammable Maths - which is in fairness not their own, they're just giving simplified content to their audience!
Their derivation is ...
- 44.7k
3
votes
0
answers
78
views
Why Stirling formula at n = -1 and n= -2 is so close to 2*pi*i [closed]
We know asymptotic formula for gamma function as Stirling formula
$$\Gamma(z+1) \approx F(z+1) = \sqrt{2\pi z}\left({\cfrac{z}{e}}\right)^z\cdot \left({1+\cfrac{1}{12z} +\cfrac{1}{288z^2}+...}\right)$$...
1
vote
1
answer
158
views
Prove or disprove that : $\int_{0}^{\infty}\left(\left(x^{\frac{3}{2}}e^{-x^{2}}\right)!-1-\frac{1}{e^{x^{2}}+1}\right)dx+\frac{7}{10}<0$
Prove or disprove that :
$$\int_{0}^{\infty}\left(\left(x^{\frac{3}{2}}e^{-x^{2}}\right)!-1-\frac{1}{e^{x^{2}}+1}\right)dx+\frac{7}{10}<0$$
It's an experiment using Desmos .
Some facts :
$$\int_{0}...
- 3,742
1
vote
0
answers
136
views
Difference between finding asymptotics of Beta function through Laplace versus through Stirling
Im trying to find the asymptotic expression for the beta function $B(x,y) = \frac{ \Gamma(x) \Gamma(y)}{\Gamma(x+y) } $. Using Stirling's approximation $\Gamma(x) \sim \sqrt{\frac{2\pi}{x} } (\frac{x}{...
- 681
1
vote
0
answers
48
views
Ask for a proof of an inequality bounding the ratio of two consecutive geometric means of the first finitely many natural numbers
A colleague asked me the first one of the following problems:
For $n\in\mathbb{N}=\{1,2,3,\dotsc\}$, is the inequality
\begin{equation}
\frac{\sqrt[n+1]{(n+1)!}}{\sqrt[n]{n!}}>1+\frac{1}{n}+\frac{\...
- 1,960
0
votes
1
answer
112
views
Inequalities about ratios of Gamma functions
Let $\Gamma(z)=\int_0^{\infty}x^{z-1}e^{-x}dx$ be the Gamma function (for the purpose of this question, one may assume $z\in\mathbb R$). I'm interested in $\Gamma(z+1/2)/\Gamma(z)$ for $z\geq 2$. From ...
- 1,319
2
votes
2
answers
67
views
Convergence of asymptotic series for $z^k/(z)_k$, $z\to\infty$
Let $z>0$, $k\in\Bbb Z$, and $(s)_n=\Gamma(s+n)/\Gamma(s)$ denote the Pochhammer symbol. According to DLMF 5.11.13 as $z\to\infty$:
$$
\frac{z^k}{(z)_k}\sim\sum_{\ell=0}^\infty\binom{-k}{\ell}B_\...
- 292
2
votes
1
answer
42
views
Asymptotic expansion of $\int \frac{1}{(y^2+c^2)^n}\, e^{-\frac{\lambda}{2} y^2} dy$
Let $\lambda>0$. Are there $c>0,K>0$ such that for all $n\in\mathbb N$
$$ \int_{-\infty}^{\infty} x^{2n}\, e^{-\frac{\lambda}{2} x^2} dx\,\ \cdot\ \int_{-\infty}^{\infty} \frac{1}{(y^2+c^2)^n}...
- 893
2
votes
1
answer
69
views
Does $\int_0^\infty \frac{s^{n-1}}{(n-1)!}e^{-s-as^b}ds$ go to zero fast?
Consider the sequence
$$a_n=\int_0^\infty \frac{s^{n-1}}{(n-1)!}e^{-g(s)}ds, \quad n \in\Bbb N,$$
where $g\colon [0,\infty)\to[0,\infty)$ is an increasing function. I'm trying to understand how its ...
- 4,289
2
votes
1
answer
180
views
Asymptotics of the incomplete gamma function
Consider the following integral
$$I=\Gamma\left(1-\alpha,\beta\right)=\int\limits _{\beta}^{+\infty}t^{-\alpha}e^{-t}dt,$$
where $\alpha,\beta>0$, and $\Gamma$ is the incomplete gamma function. Is ...
