Linked Questions

42
votes
5answers
135k views

Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do you consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? Thanks.
47
votes
3answers
3k views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
11
votes
7answers
1k views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
9
votes
2answers
693 views

Equality with Euler–Mascheroni constant

While trying to prove $\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show? in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log ...
6
votes
3answers
612 views

Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...
6
votes
4answers
154 views

Asymptotic approximation regarding the Gamma function $\Gamma$.

On the wikipedia page for Gamma function I saw an interesting formula $$ \lim_{n\to \infty} \frac{\Gamma(n+\alpha)}{\Gamma(n)n^\alpha} = 1 $$ for all $\alpha\in\Bbb C$. I couldn't find the source of ...
5
votes
2answers
246 views

Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
4
votes
3answers
159 views

Evaluate: $\lim\limits_{r \to \infty} \frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}$

Evaluate: $$\lim\limits_{r \to \infty} \frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}$$ My effort: \begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \...
2
votes
2answers
258 views

a proof for the following Gamma function inequality

Could you please provide or point me to a proof of inequality 5.6.8 found at this site? That is, $\left|\frac{\Gamma(z+a)}{\Gamma(z+b)}\right| \leq \frac{1}{|z|^{b-a}}$ for $z\in \mathbb{C}$, $a,b\...
3
votes
1answer
82 views

How to show that $\frac{\sqrt{n-1}\Gamma((n-1)/2)}{\sqrt{2}\Gamma(n/2)}>1$.

Show that $$\frac{\sqrt{n-1}\Gamma((n-1)/2)}{\sqrt{2}\Gamma(n/2)}>1$$ I tried to solve it using Taylor series expansion.
1
vote
1answer
100 views

Confirm result of $2^n\int_{0}^{1}{x^{2^n-1} \over 1+x^{2^n}}\ln{(-\ln{x})}\mathrm dx=-{2n+1\over 2}\cdot\ln^2{2}$

Consider the integral $$2^n\int_{0}^{1}{x^{2^n-1} \over 1+x^{2^n}}\ln{(-\ln{x})}\mathrm dx=-{2n+1\over 2}\cdot\ln^2{2}\tag1$$ $n\ge0$ An attempt: $u=x^{2^n}$ $\implies$ $du=2^n\cdot x^{2^n-1}dx$...
2
votes
1answer
84 views

Weierstrass Factorization theorem on the Reciprocal Gamma Function

I am just a bit curious about the Weierstrass Factorization theorem on the Gamma function. The Weierstrass Factorization theorem says this: Let $f(z)$ be an entire function. Suppose that $f$ ...