Linked Questions

4
votes
4answers
411 views

$\left(-\frac{1}{2}\right)! = \sqrt{\pi}?$ [duplicate]

I recently learned that $\left(-\frac{1}{2}\right)! = \sqrt{\pi}$ but I don't understand how that makes sense. Can someone please explain how this is possible? Thanks!
3
votes
2answers
109 views

$\frac{1}{2}!$ aka $\Gamma(\frac{3}{2})$ [duplicate]

I know it's $\frac{\sqrt{\pi}}{2}$ but how can this be evaluated by hand? (Or can it not?) For quick reference: $$n!=\Gamma(n+1)$$ $$\Gamma(n)=(n-1)!$$ $$\Gamma(n)=\int_0^\infty x^{n-1} e^{-x}\,dx$$ ...
171
votes
19answers
49k views

Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
86
votes
10answers
6k views

Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: $$\...
15
votes
3answers
813 views

Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$

It can be shown that using the definition of the Gamma function as: $$\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx $$ that $$\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$$ or slightly abusing notation, that $(-\...
8
votes
4answers
422 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
0
votes
1answer
3k views

Evaluating the Gamma function

So I found out about the gamma function yesterday and I spent a bunch of time trying to evaluate certain values like $0.5!=\Gamma \left(1.5\right)$. I used multiple integration by parts, and in the ...
3
votes
1answer
546 views

Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. Also,...
3
votes
3answers
189 views

Which methods can be used to evaluate the following integral?

How can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$.
0
votes
4answers
68 views

Evaluate $\lim_{n \to \infty}\int_{0}^{\infty}e^{-nx}x^{-1/2}dx$

I would like to evaluate $$\lim_{n \to \infty}\int_{0}^{\infty}e^{-nx}x^{-1/2}dx.$$ The purpose of the problem was to show that (1) $\int_{0}^{\infty}e^{-nx}x^{-1/2}dx$ converges for every natural ...
1
vote
1answer
278 views

applications of euler's reflection formula.

In this post,one of the answers (in fact the answer with more upvotes) uses euler's reflection formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi\,z)}$ for the gamma function $\Gamma(z)$ to evaluate ...
2
votes
1answer
97 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
1
vote
3answers
77 views

Where did I go wrong in proving $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$

Let $X$ ~ $\mathcal{N}(0,1)$ Show that: $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$ Idea: $\mathbb E[X^{2n}]=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}x^{2n}e^{-\frac{x^2}{2}}...
5
votes
1answer
127 views

Not-too-slow computation of Euler products / singular series

I'd like to compute, to at least a few digits of accuracy, the constants that arise in Hardy-Littlewood conjecture F / Bateman-Horn conjecture, in particular for just a single quadratic polynomial. ...
8
votes
0answers
150 views

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually?

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$) If there is/are, could you show me how to calculate it? I found that $\Gamma(...

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