Questions tagged [gamma-function]
Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.
3,308 questions
1 vote
1 answer
119 views
On $\int_0^\infty\left[\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^s}{(e^x-1)^2}\right]dx$
Calculate: $${\int\limits_0^\infty{\left[{\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^{s}}{(e^x-1)^2}}\right]dx=}{\,\,\color{red}{?}}}\tag{1}$$ For $0\lt\Re(s)\lt1$. An integral definition of $\zeta(s)$ Zeta ...
- 5,102
2 votes
0 answers
98 views
Is this proof of the Gaussian Integral valid?
I came across this method of proving the Gaussian Integral when messing around with it. More precisely, I found a way to prove the integral $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar ...
- 29
3 votes
1 answer
75 views
Airy zeta as a polynomial
The Airy zeta function is defined as a sum over the zeroes of the $\mathrm{Ai}$ airy function. $$\zeta _{\mathrm {Ai} }(s)=\sum _{i=1}^{\infty }{\frac {1}{|a_{i}|^{s}}}$$ Integer values for $\zeta_{\...
- 5,205
-3 votes
2 answers
103 views
Does this series $\sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(\frac{-n}{2})}$ converge? [closed]
In the series below, I'm aware that dividing a number by infinity yields zero, and I'm curious why Mathematica can't solve it. $$ \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(\frac{-n}{2})} $$
- 1,988
2 votes
0 answers
58 views
Intervals of continuity of $ f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1}$
Let $$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt.$$ Define $$f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1},\qquad x\in[0,\infty).$$ Find the maximal open intervals $I_n=(a_n,b_n)\subset[0,\infty)...
- 64
-3 votes
4 answers
315 views
Is the integral $\int_0^{\infty} \cos x \ln \left(1+x^2\right) d x$ convergent?
Latest Edit Being reminded that the integral is divergent, I open the discussion on the convergence of the integral. Thank all of you who raised the question for discussion. I shall show my recent ...
- 33.2k
15 votes
8 answers
640 views
Square root gamma function inequality $\sqrt{\Gamma(x)} \ge \Gamma(\sqrt{x})$ for $x > 0$
While playing with random functions and graphs on desmos, my friend showed me a plot of a $\sqrt{\Gamma(x)}$ and $\Gamma(\sqrt{x})$,what I found interesting is that $\sqrt{\Gamma(x)}$ is always bigger ...
- 357
3 votes
3 answers
507 views
Reducing an integral to a Gamma function type
I would to prove that: $$ \bbox[5px,border:2px solid #F5B041]{\int_0^1 x^m (\ln x)^n \, dx = (-1)^n \cdot \frac{n!}{(m+1)^{n+1}}, \quad \text{for } m > -1, \, n \in \mathbb{N}.} $$ Since $\ln x$ ...
- 8,886