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

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3
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0answers
62 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
47 views
+100

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
53 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
20 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
38 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
26 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
54 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
97 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
60 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
93 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
73 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 ...
1
vote
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
38 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 ...
1
vote
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
45 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
43 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
65 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} = ...
1
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 ...
1
vote
1answer
71 views

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

Can I solve this equation $$\binom{4}{x}^2-\binom{4}{x}-25=0$$
0
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}+ ...
3
votes
2answers
44 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
0
votes
1answer
18 views

Gamma function identity used in deriving negative binomial from gamma-poisson mixture

On this wikipedia page negative binomial distribution, Negative binomial was derived as integrating out the lambda from Gamma-Poisson mixture. I tried to follow the proof step by step, but I am stuck ...
0
votes
1answer
53 views

How to prove this gamma function identity?

Reading Landau and Lifshitz "Quantum Mechanics. Non-relativistic theory", I've come across an identity, which after being a bit simplified, reads ...