# Tagged Questions

87 views

### Compute $\Gamma(2.7)$

All I know of the Gamma function is $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$ and the recursive formula $$\Gamma(\alpha) = (\alpha-1)\Gamma(\alpha - 1)\text{.}$$ A ...
98 views

### Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
12 views

### Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $\gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
27 views

### Convergence of $\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$

This question is a place to store proofs of the convergence of Euler's product formula for the gamma function: $$\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$$ which is convergent for ...
68 views

### A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
48 views

### Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0$ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
51 views

### Partial Fractions Decomposition of the Gamma Function

I'm currently dealing with a problem my professor raised (since I just studied the Mittag-Leffler's Partial Fractions Theorem). The problem is to derive a partial fractions decomposition of the Gamma ...
94 views

### Derivative of $\Gamma$ at $1$

I've been given two definitions of the Gamma function, the integral defintion: $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt$ (for $Re(z)>0$) and the product definition (for $1/\Gamma$): ...
33 views

### Laguerre polynomial defined as contour integral

Let $P_n(x)=\frac{1}{2\pi i}\oint_{\Sigma}{\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^tdt}$ be a polynomial of degree $n$ where $n\in\mathbb{N}, x>0$ and $\Sigma$ is a closed contour in the $t$-plane that ...
54 views

### Is there a simpler form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$?

Is there a simpler (i.e. manifestly real) form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$ or $\Im \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$, or more generally for $\frac{\Gamma(1/2-ia)}{\Gamma(1-ia)}$ with ...
181 views

### Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0.$$ I am ...
38 views

### factorization of an expression involving gamma function

Does the equation $\Gamma(x+1/2)\Gamma(x-1/2)=\Gamma(x+iy)\Gamma(x-iy)$, where $\Gamma(z)$ is the Gamma function and $i=\sqrt{-1}$, have any solution assuming $x,y$ are both real and $x>1/2$? This ...
60 views

### Can the hypergeometric function be extended analytically to the complex plane in the interval [1,$\infty$ )?

Just a thought. The hypergeometric function, which can be written as: $$F(a,b,c \space;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$$ is obviously ...
310 views

### Ahlfors “Prove the formula of Gauss”

He says: Prove the formula of Gauss: $$(2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n})$$ This is an exercise out of Ahlfors. ...
33 views

### Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
145 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 ...
87 views

### Find a formula for $\Gamma(\frac{n}{2})$ for positive integer n.

Find a formula for $\Gamma(\frac{n}{2})$ for positive integer n. I know the following relations; $\Gamma (z+1)=z\Gamma (z)$ and $\Gamma(n+1)=n!$ Please give me a way how to show this. Thank ...
106 views

106 views

### Calculation of $\Gamma(n+\frac12)$

I want to show that $$\Gamma\left(n+\dfrac12\right)=\dfrac{1\cdot 3\cdot 5\cdots(2n-1)}{2^n}\sqrt{\pi}=2^{1-2n}\dfrac{\Gamma(2n)}{\Gamma(n)}\sqrt{\pi}.$$ I know that $\Gamma(\frac12)=\sqrt{\pi}$. The ...
40 views

### Magnitude of $\Gamma(1/2+it)$

I want to prove that $$|\Gamma(1/2+it)|=\sqrt{\frac{2\pi}{e^{\pi t}+e^{-\pi t}}}$$ My idea is to use the formula $\Gamma(s)\Gamma(1-s)=\pi/\sin \pi s$. Plugging in $s=1/2+it$ and taking absolute ...
272 views

### Gamma function has no zeros

I want to show that $\Gamma(z)$ has no zeros. My idea is to use the formula $$\Gamma(z)\Gamma(1-z)=\dfrac{\pi}{\sin(\pi z)}$$ which holds for all $z\in\mathbb{C}$. If $\Gamma(z)=0$, then the ...
55 views

### Factorial and product form of gamma function

Let $z\in\mathbb{C}$. We know that $$\dfrac{1}{\Gamma(z)}=e^{\gamma z}\prod_{n=1}^\infty\left(1+\dfrac{z}{n}\right)e^{-z/n},$$ where $$\gamma=\lim_{N\rightarrow\infty}\sum_{n=1}^N\dfrac1n-\log N.$$ ...
126 views

### $\Gamma(1/2-n+it)$ converges uniformly

Prove that $\Gamma(1/2-n+it)\rightarrow 0$ uniformly as $n\rightarrow\infty$ for $t\in\mathbb{R}$, where $n$ is a positive integer. I'm not sure which definition of $\Gamma$ would be easiest to ...
### $| \Gamma (x_{n})| \to 0$ as $n \to \infty$ where $x_{n} \in [-n+\delta,1-n-\delta]$
I'm trying to prove $| \Gamma (x_{n})| \to 0$ as $n \to \infty$ where $x_{n} \in [-n+\delta,1-n-\delta]$ for some $\delta > 0$. What possible method can I use? As a special case, I've proved ...