3
votes
1answer
48 views

Prove $\lim_{n \to \infty} \frac{\Gamma(n+1/2)}{\Gamma(n)~n^{1/2}}=1$

Prove $$\lim_{x \to \infty} \frac{\Gamma(x+1/2)}{\Gamma(x)~x^{1/2}}=1.$$ I got this problem from Probability and Statistics by Degroot & Schervish. There is a hint to use Stirling's formula ...
1
vote
1answer
101 views

Expansion of lower incomplete gamma function $\gamma(s,x)$ for $s < 0$.

The lower incomplete gamma function for positive $s$ is defined by the integral $$ \gamma(s,x)=\int_0^{x} t^{s-1} e^{-t} dt. $$ Taylor expansion of the exponential function and term by term ...
0
votes
1answer
65 views

Asymptotes of $\Gamma(\frac{1}{2} +ix)$ when $\vert x \vert \to \infty$

I am currently looking for finding behaviour of the function $\vert \Gamma(\frac{1}{2}+ix) \vert$ when $x$ tends to $\infty$. I think I need to use the Stirling's approximation but I don't see how. ...
1
vote
0answers
41 views

Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
10
votes
3answers
265 views

Estimate $\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt$ accurately.

How can I obtain good asymptotics for $$\gamma_n=\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt\text{ ? }$$ [This has been already done] In particular, I would like to obtain asymptotics that ...
1
vote
1answer
105 views

Asymptotic formula for complex gamma function at $+\infty+i \times y$

I am currently looking for the behaviour of the complex gamma function at real infinity: $\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$ and more particularly for asymptotic formulas for the ...
2
votes
3answers
69 views

Asymptotic behaviour of $1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$

I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of $$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} ...
3
votes
1answer
98 views

Behavior of $\Gamma(z)$ as $\text{Im} (z) \to \pm \infty$

In a paper I'm reading it states that $\displaystyle |\Gamma(z)| = |\Gamma(a+ib)| \sim \sqrt{2 \pi} |b|^{a-\frac{1}{2}} e^{-|b|\frac{\pi}{2}}$ as $\displaystyle|b| \to \infty$. How is that derived ...
1
vote
0answers
138 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
0
votes
1answer
137 views

Upper bound of function including Pochhammer symbol

How can I find the upper bound of $$\left\vert\frac{(c+1/2+\lambda)_{n}}{\lambda^{n}}\right\vert,\quad\text{where}\quad(c+1/2+\lambda)_{n}=\frac{\Gamma(c+1/2+\lambda+n)}{\Gamma(c+1/2+\lambda)}$$ and ...
2
votes
0answers
59 views

gamma funtion and estimates-typo or mistake?

In one of the lecture notes I've found that $C_n$ $$ C_n= \begin{cases} \frac{n!}{\sqrt 2 \Gamma((n/2+1)}\pi^{-1/42^{-n/2}(n!)^{-1/2}} & n\text{ even} \\[4mm] \frac{2(n!)}{(\sqrt2n+1/(\sqrt2 ...
14
votes
6answers
3k views

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = ...
5
votes
3answers
627 views

Asymptotics of terms and errors in Stirling's Approximation

I have two related questions. Both are related to the asymptotics of Stirling's approximation, which is why I have included them in the same question. I will separate the questions if it is deemed ...
7
votes
1answer
524 views

the limit of the ratio of two $\Gamma(x)$ functions

I am interested in the quantity $$ a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$$ (this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch ...
4
votes
1answer
662 views

Variations on the Stirling's formula for $\Gamma(z)$

I am currently reading some material that makes heavy usage of Hypergeometric functions, and there is one particular point about applying Stirling's approximation to various terms consisting of ...