Linked Questions

5
votes
3answers
2k views

Why do we use 'this' Gamma Function. [duplicate]

The Gamma function is a generalization of the factorial defined by Euler as: $$\Gamma(z)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}\,dt$$ for $z\in\mathbb{C}$ with positive real part. It satisfies, $\...
2
votes
2answers
142 views

Is the gamma function $\Gamma(n+1)$ the only continuous function and defined derivative with the same recursive definition as of $n!$ for $n>-1$? [duplicate]

Is the gamma function $\Gamma(n+1)$ the only continuous function and defined derivative with the same recursive definition as of $n!$ for $n>-1$ ? (When using real numbers.) The recursive ...
-2
votes
1answer
161 views

Why is the Gamma function $\Gamma$ used for interpolating factorials? Is it wrong to say that $(4.3)! = \Gamma(3.3)$? [duplicate]

I could not understand that whyGAMMA function is used for interpolation? Isn't it wrong to say that that factorial of 4.3 is gamma 3.3?
0
votes
1answer
61 views

What does $n!$ mean if $n$ is not an integer? [duplicate]

I was always taught that $n!$ simple meant multiplying together all numbers up to and including $n$. So for example, $3!$ meant $3 \times 2 \times 1$. Now I discover that $n$ does not have to be an ...
1
vote
0answers
45 views

Where does the gamma function come from [duplicate]

It has long been known that the gamma function is an extension (shift) of the factorial function defined on integers. There were an infinite numbers of ways to continue the factorial, so what ...
188
votes
6answers
11k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
73
votes
3answers
6k views

Why is $i! = 0.498015668 - 0.154949828i$?

While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!. Curiously, Google's ...
25
votes
3answers
13k views

How to come up with the gamma function?

It always puzzles me, how the Gamma function's inventor came up with its definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
5
votes
3answers
3k views

Estimating the Gamma function to high precision efficiently?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma ...
32
votes
2answers
2k views

Explicitly reconstructing a function from its moments

Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge absolutely....
4
votes
2answers
2k views

Domain of the Gamma function

I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral: $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$ converges. I started by ...
0
votes
5answers
434 views

Can the value of $(-9!)$ be found [duplicate]

I saw this question on an fb page and I couldn't solve it. Question: What is the value of $(-9!)$? a)$362800$ b)$-362800$ c) Can not be calculated The first options seems to be incorrect,which ...
11
votes
1answer
1k views

The uniqueness of the Gamma Function

It is a theorem that any function $f$ defined for positive real numbers satisfying $f(1)=1$ $f(x+1)=x\cdot f(x)$ $f$ is log convex is identically equal to the gamma function. (Condition 2 means that ...
3
votes
2answers
472 views

Newton's Interpolation formula (series) on the factorial

I was wondering what would happen if we took Newton's Interpolation formula (series) and applied it the factorial. It is given as $$f(x)=\sum_{k=0}^\infty\binom{x-a}k\Delta^k[f](a)$$ where $$\...
4
votes
2answers
586 views

Why do many calculators evaluate $(-0.5)!$ to $\sqrt\pi$?

According to Wikipedia, factorial only is defined for non-negative integers. How come Spotlight, the Windows calculator and the Google search engine come up with $\sqrt\pi$ if you try to solve $(-0.5)!...

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