2
votes
2answers
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 ...
7
votes
2answers
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, ...
0
votes
0answers
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]. ...
2
votes
2answers
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 ...
7
votes
1answer
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 ...
2
votes
0answers
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 $\forall s \in \mathbb C$, ...
3
votes
0answers
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 ...
2
votes
2answers
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 $1/\Gamma$): ...
0
votes
0answers
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 ...
3
votes
1answer
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 ...
7
votes
4answers
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
5
votes
2answers
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. ...
0
votes
0answers
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: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
1
vote
1answer
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 ...
1
vote
2answers
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 ...
2
votes
1answer
106 views

Prove that $\Gamma'(1)=-\gamma$

Use the product formula for $1/\Gamma(z)$ to prove that $$\Gamma'(1)=-\gamma$$ I know that for Euler constant $\gamma$, $$\frac{1}{\Gamma(z)} =ze^{\gamma z}\prod _{k=1}^{\infty} ...
1
vote
4answers
212 views

The complex gamma function

Show that $$\Gamma (z+1)=z\Gamma (z)$$ $\forall z\in \Bbb C$ except for $z=-n$ where $n\in \Bbb N$. I know that the gamma function is defined as $\Gamma (z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$ And ...
1
vote
2answers
154 views

Gamma function in complex analysis.

Prove that $$ \Gamma\left(z\right) = \lim_{n\to \infty}\int_{0}^{n}t^{z - 1}\left(1 - {t \over n}\right)^{n}\,{\rm d}t \quad\mbox{for}\quad \Re z \gt 0 $$ I know that $$ {\rm e}^{-t/n} = 1 - {t ...
5
votes
2answers
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 ...
4
votes
2answers
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 $$\gamma=\lim_{N\rightarrow\infty}\sum_{n=1}^N\dfrac1n-\log N.$$ ...
7
votes
1answer
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 ...
1
vote
0answers
117 views

Euler reflection formula via Wielandt theorem

I would like to prove Euler's reflection formula $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$ using Wielandt's theorem: Let $f$ be a function that is bounded on the strip $1 \le ...
1
vote
1answer
92 views

Formula for $\Gamma (\frac{1}{2} + i t)$

I have been working on the following problem for my complex analysis class involving Euler's Gamma function: For $$\Gamma (s) := \int_0 ^{\infty} t^{s-1} e^{-t} \,dt \ , \ Re(s)>0$$ Show that ...
1
vote
1answer
30 views

path of the integral in the initial definition of gamma function

Can the path of the integral in the initial definition of gamma function be altered to a straight line starting from $0$ to $\infty;e^{ia},a<\pi/2$)?
0
votes
0answers
74 views

Integral formula for Gamma function over rays other than $(0,\infty)$

I would like to understand the identity $$ \lambda^{-z}\,\Gamma(z) = \int^\infty_0 t^{z - 1} e^{-\lambda t} \,dt\,,\qquad \text{valid for } \Re(z) > 0, \Re (\lambda) > 0. $$ Here the complex ...
6
votes
1answer
113 views

Proof of $\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$

I am trying to prove the identity $$\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$$ for $0 < \Re(z) < 1$, starting from the integral definition of the gamma function $$\Gamma(z) = ...
22
votes
2answers
507 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
9
votes
2answers
298 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. Specifically, ...
5
votes
1answer
155 views

A gamma function identity

I am given the impression that the following is true (for at least all positive $\lambda$ - may be even true for any complex $\lambda$) $$ \left\lvert \frac{\Gamma(i\lambda + 1/2)}{\Gamma(i\lambda)} ...
1
vote
1answer
107 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 ...
4
votes
2answers
158 views

How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?

(Source: Wolfram Alpha) Or, to write it out in full, $$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$ How is this identity derived? Also, knowing this, could we find the exact values for the real ...
1
vote
1answer
129 views

Limits of complex error and gamma functions in the complex plane?

What are the following one-sided limits in the complex plane (in the form $x+iy$): For the complex error function: $\lim_{x \to 0^+, y \to 0^+}\text{erf}\left(x+iy\right) = $ $\lim_{x \to +\infty, ...
4
votes
1answer
131 views

Integral formula for $\frac{1}{\Gamma(z)}$

Let $c>0$. How to prove that for any complex number $z$, $$\frac{1}{\Gamma(z)}=\frac{1}{2\pi}\int_{-\infty}^\infty (c+it)^{-z}e^{c+it}\,dt?$$ where $\Gamma(z)$ is the Gamma function.
1
vote
1answer
168 views

Generating Laguerre polynomials using gamma functions

An exercise given by my complex analysis assistant goes as follows: For $n \in \mathbb{N}$ and $x>0$ we define $$P_n(x) = \frac{1}{2\pi i} \oint_\Sigma ...
1
vote
0answers
79 views

Gamma Function Problem

Hi is it fair to write $$\Gamma(1+ix)=ix\Gamma(ix) $$ and then to say that $\Gamma(ix)=\exp(i\arg(\Gamma(x)))$. If not can anyone explain what $\arg(\Gamma(x))$ is defined as please, I always get ...
1
vote
1answer
277 views

half-line Fourier transform of $x^{z-1}$ w.r.t. $x$?

Can someone help me evaluate $G_g(z)=\int_0^{\infty}x^{z-1}e^{igx}dx$, where $g$ is real and $z$ is complex? By closing the contour in the upper half plane, I've managed to prove that if ...
1
vote
0answers
58 views

What is the closed form expression for this?

Let $r_1...r_k$ be the $k$ roots unity or solutions to the expression $x^k = 1$ What is the expression: ...
3
votes
0answers
148 views

Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
1answer
339 views

how to calculate gamma function in programming language? [duplicate]

how to calculate gamma function in programming language? I also need to support complex numbers (complex numbers are in the programming language) and negative numbers. thank you :)
-1
votes
1answer
81 views

Is there any meaning on $\displaystyle\Gamma(i)$ [closed]

Is there any meaning on the $\displaystyle\Gamma(i)$ ?
4
votes
1answer
106 views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ I don't know how to do this ? Note that $\gamma $ is the Euler-Mascheroni constant
5
votes
2answers
120 views

Computing $\lim_{s \to 1} \Gamma \left(\frac{1-s}{2}\right) (s-1)$

I want to evaluate the following limit: $$\lim_{s \to 1}\; \Gamma \left( \frac{1-s}{2} \right) (s-1).$$ I know that the gamma function has simple poles at $-n$ for $n \in \mathbb{N}_0$ with residue ...
0
votes
1answer
75 views

Question on Gamma Function

In Gelfand and Shilov Vol I (of Generalized Function), on page 257, they write down the following equation that I don't know how to arrive at: $$\int_{0}^{1} (1-t)^{-\frac{n}{2}} t^{\frac{q-2}{2}}dt ...
4
votes
3answers
371 views

Limiting behavior of gamma function

I am trying to determine whether $\Gamma(x+iy)\rightarrow 0$ as $y\rightarrow\infty$. How should I go about doing it? I was trying to see if I could get anything from ...
0
votes
2answers
60 views

Sequential Limit of Line Integral Is The Same As The Usual Limit of Line Integral? (Gamma Function Related)

Let $\epsilon > 0$, and $ n \in \mathbb{Z}^{+} $. Let $C_{n}$ be a positively oriented polygonal line that is from $-n + 1/2 - i \epsilon$ to $ 1/2 - i \epsilon$ and from $ 1/2 - i \epsilon$ to $ ...
0
votes
2answers
53 views

$| \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 ...