Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

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2
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1answer
68 views

Divergent products.

My question are about divergent products. I'm a Dutch student so i may lack the skil to write it down in the correct notation and forgive my spelling errors. A thing i've found on the internet was ...
5
votes
2answers
30 views

The Duplication Formula for the Gamma Function by logarithmic derivatives.

I was reading Ahlfors' "Complex Analysis" (second edition) and in Chapter 5, section 2.4, where he studies the Gamma Function, he proves Legendre's Duplication Formula: ...
0
votes
1answer
10 views

Gamma distribution - closed towards multiplication

First observe how the gamma distribution function can be written in terms of the incomplete gamma function. $\boldsymbol{(1)} \qquad G(y) = \int_{0}^{y} \dfrac{c^{\gamma}}{\Gamma(\gamma)} x^{\gamma - ...
4
votes
1answer
85 views

Trying to understand a step in Ramanujan's Proof of Bertrand's Postulate regarding the gamma function

My question relates to this step in the proof here: But it is easy to see that $$\log \Gamma(x)-2\log\Gamma(\frac12x+\frac12) \le \log\left\lfloor ...
3
votes
1answer
54 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
1
vote
1answer
56 views

How can I prove this integration result?

The question is: how can I prove that: $$\int_{0}^{\pi} \sin^n\theta\ d\theta = \frac{\Gamma\big(\frac{1}{2}\big) \Gamma\big(\frac{1}{2} + \frac{1}{2}n\big)}{\Gamma\big(1 + \frac{1}{2}n\big)}$$
2
votes
2answers
54 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
4
votes
3answers
45 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
0
votes
1answer
32 views

Some problem with Euler's Reflection Formula?

$$\Gamma(z)\Gamma(1-z)=\int_0^{\infty}x^{z-1}e^{-x}dx\int_0^{\infty}x^{-z}e^{-x}dx=\int_0^{\infty}(2x)^{-1}e^{-2x}(2dx)=\Gamma(0)\to\infty??$$
3
votes
1answer
67 views

Proving $\Gamma(x)$ is holomorphic

My professor defined Gamma function in the following way:$$\Gamma(z)= \lim \limits_{n \rightarrow \infty} \frac{n!n^z}{z(z+1)....(z+n)}$$ Now we first observe that $f_n(z)= ...
2
votes
1answer
43 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...
1
vote
0answers
34 views

What is the inverse function of the Log Gamma function?

What is the inverse function of the Log Gamma function? $\log\Gamma(x)$ http://mathworld.wolfram.com/LogGammaFunction.html Can it be inverted, and why not, if not?
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vote
0answers
21 views

Infinite product for a generalization of gamma function

Is there a generalization of Weierstrass infinite product to the function $$ \frac{1}{g(x,a)}=xa^{-x}e^{\Psi (a) x}\prod_{n=0}^{\infty}\left(1-\frac{x}{c_{n}(a)}\right)e^{\frac{x}{n+a}}$$ $ \Psi(a) ...
2
votes
3answers
59 views

Simplify ${\Gamma(\beta)\Gamma(u) \over \Gamma(\beta+u)}$ [closed]

Is it possible that the expression below be simplified to a simpler form: $${\Gamma(\beta)\Gamma(u) \over \Gamma(\beta+u)}$$ whereby $\beta$ is a variable
0
votes
1answer
39 views

Simplify Infinite Series Involving Gamma Function $\Gamma$

This question originated from this problem. Can anyone help me simplifying the infinite series below: $$\sum_{n=0}^{+\infty}\frac{1}{\Gamma(\beta_2-n)}\frac{(-e^{-t})^n}{n!}$$ The only idea I have ...
0
votes
1answer
27 views

A Fourier series' upper bound involving gamma function

I am reading Donald E. Knuth's "The Art of Computer Programming" Vol. 3 and stuck on the equation 47 and inequality 48 on Page 133,which are the follows: $$ \delta(n)=\frac{2}{\ln2}\sum_{k \ge 1} ...
1
vote
2answers
55 views

Why does $\Gamma(1-z)\Gamma(z) = \frac{\pi}{\sin \pi z}$ imply $\Gamma(z) \not = 0$

I'm reading on the extension of $\Gamma$ to the complex plane and there is written: Corollary $$\Gamma(z) \not = 0 \qquad \forall z \in \mathbb{C}\setminus\{0,-1,-2, \dots\}$$ Proof ...
1
vote
3answers
98 views

Recurrence Relation Involving the gamma Function

I'm having some doubts about my approach to the following problem. I am given that the function $k(z)$ is defined such that, ...
1
vote
0answers
61 views

Inequality involving integral of $\Gamma(x)$

The graph of $\frac{e^x}{\Gamma(x+1)}$ is somewhat bell-shaped. I think the proof of the following requires an understanding of the integral of this function that I can't glean from either Mathematica ...
0
votes
1answer
79 views

Proof of a formula involving Gamma function

I am trying to prove the following formula for $\operatorname{Re} z>0$ 􀀀$$\frac{\Gamma(z)\sin\theta z}{n(a^2+b^2)^{z/2}} = \int_0^\infty e^{-at^n}t^{nz-1} \sin(bt^n)\,dt$$ where $n$ is a ...
0
votes
1answer
17 views

$\Gamma(p) \in C^2(\mathbb{R})?$

I'm trying to show that the gamma function is twice continuously differentiable for $p>0$. I was wondering whether the following is actually valid? Or is the simply the way of computing the ...
6
votes
1answer
94 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
3
votes
0answers
74 views

What is $\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$?

Consider the infinitely nested expression $$x=\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$$ where $\Gamma$ is the Gamma function. Imitating the standard method for solving infinitely nested radicals, we ...
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0answers
31 views

Sum of all n dimensional spheres?

I was messing around and made some code to find the area of an n dimensional sphere. I noticed that as n increases, the area tends towards zero. These were the results: ...
1
vote
1answer
39 views

Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
0
votes
1answer
24 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
1
vote
3answers
96 views

Value of $\sum_{n=0}^{\infty} \exp(-bn^a)$

What is the value of $$ S = \sum_{n=0}^{\infty} \exp(-bn^a) $$ where $a>0$ and $b>0$? I know that $$ \int_0^{\infty} \exp(-bx^a) \, \mathrm{d}x=\frac{1}{ab^{1/a}}\Gamma(1/a) $$ but cannot ...
1
vote
1answer
139 views

Help to solve complex equation related to the Gamma function

I would need some help to solve the next complex equation for $y\in\mathbb {R}$, which I already know to be real-valued: $$ \frac {1} {2i}\left ((2\pi)^{\text {iy}}\text {}\text {Sin}\left (\frac ...
1
vote
1answer
29 views

inequality regarding the gamma function

I am curious to know if the following is true: Let $\alpha >0$. Then \begin{equation} \int_0^K x^{\alpha -1} e^{-x} \,dx \leq \Gamma (\alpha), \quad \forall K \in \mathbb{R}. \end{equation} This ...
2
votes
3answers
84 views

Evaluating $\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx$.

I have found this result on the Internet $$\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx = \Gamma(n+1).$$ I know that if $n \in \mathbb{N}$, the proof is not complicated. However, if $n \in ...
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0answers
20 views

Proving a bound for $\Gamma(s+u)/\Gamma(s)$

Suppose $s, w$ are complex numbers with positive real part. I have come across a particular bound that I have seen multiple times, but which I do not know how to prove: ...
1
vote
1answer
46 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...
0
votes
0answers
44 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
1
vote
1answer
66 views

Zeta and Gamma function regularization with $\omega=1/0$

I have recently read about zeta function regularization, a way of ascribing values to functions having simple poles in a point and to divergent series. The values obtained are the same as those ...
3
votes
0answers
28 views

property of complex gamma function

Show that $$|(n+ib)!|= \left( \frac{\pi b}{\sinh(\pi b)}\right)^{{1}/{2}}\prod_{s=1}^{n}{(s^2+b^2)^{{1}/{2}}}$$ I have the following relations $$\frac{\sinh(b \pi)}{b\pi} = ...
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vote
0answers
35 views

How to compute this limit involving Gamma functions

I am investigating the following limit $$\lim_{u\to \infty}\frac{\Gamma \left(\frac{1}{2 H},\frac{H 2^{\left(\frac{1}{2H}\right)}}{u^{\frac{1}{H}}\left(1-H\right)^{\left(2 - \frac{1}{H}\right)}} ...
0
votes
0answers
18 views

Upper bound of $\int_{z_1}^{z_2}x^k e^{-x}dx$

How can I find a good upper bound to this quantity ? $$\int_{z_1}^{z_2}x^k e^{-x}dx,\ \ \ \ z_{1,2}\in \mathbb {Im}, k\in \mathbb N$$ The integrand makes me thinking about an incomplete gamma ...
3
votes
1answer
41 views

Closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$?

I need the closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$? The title is fairly self-explanatory. I know that a closed form of this exists using the ...
0
votes
2answers
36 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...
0
votes
3answers
78 views

Integral with $e$ , $ \int e^{x^3}\;dx$ [duplicate]

$ \int e^{x^3}\;dx$ so, i'm searching for answer this question, i think that this not is too easily, but i think this integral not exist solution undefined, this integral would be easy if had the ...
1
vote
1answer
29 views

Stirling formula on $\frac{\gamma(\frac{1}{2}s)}{\gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I tried to prove $\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ using stirling formula when $s =\sigma+it$. However, since stirling formular for gamma function is ...
9
votes
3answers
128 views

Evaluating this integral using the Gamma function

I was wondering if the following integral is able to be evaluated using the Gamma Function. $$\int_0^{\infty}t^{-\frac{1}{2}}\mathrm{exp}\left[-a\left(t+t^{-1}\right)\right]\,dt$$ I already have a ...
1
vote
1answer
34 views

Gamma of 3z using triplication formula:

I have to demostrate the gamma function for 3z as you see below: Using the multiplication formula demostrate gamma(3z) Gamma functions of argument $3z$ can be expressed using a triplication ...
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1answer
71 views

Solve this equation $\binom{4}{x}^2-\binom{4}{x}-25=0$

Can I solve this equation $$\binom{4}{x}^2-\binom{4}{x}-25=0$$
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votes
0answers
44 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
1
vote
1answer
34 views

Log of 1-reguralized incomplete gamma function (upper)

I must compute value of 1-regularized incomplete gamma function (upper) $Q(a,z)$. But unfortunatelly this computing exceeding the precision of the processor (for example I gain 0, but I should have ...
5
votes
1answer
80 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
0
votes
2answers
40 views

Euler and the factorial function

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that ...
7
votes
2answers
77 views

How does $\int_0^\infty e^{-t^4}dt = \Gamma (\frac{5}{4}) ?$

My text book claims that $$\int_0^\infty e^{-t^4}dt = \Gamma \left(\frac{5}{4}\right).$$ I fail to see this. By the definition of the gamma function we have $$\Gamma (z) = \int_0^\infty ...
1
vote
1answer
56 views

Integrate $\frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$ with respect to $y$

$$F(x,y)= \frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$$ Please show that the function when integrated with respect to $y$ is $F_X(x)= \frac{\lambda}{(2\sqrt{2}(\frac{1}{2}+ ...