3
votes
How to prove that $\big(\operatorname{sinc}(x) + \operatorname{sinc}(x-1) \big)^{*r} = \frac{\Gamma{(r+1)}}{\Gamma{(x+1)}\Gamma{(r-x+1)}}$
First of all, recall that the $\mathrm{sinc}$-function is defined to be
$$\mathrm{sinc}(x) := \frac{\sin{(\pi x)}}{\pi x}$$
and is well-defined also at the origin. Next thing is to note that the $\...
3
votes
Integral of Choose functions
Just use the definition of the generalized binomial coefficient:
$$\int_{0}^{4}\binom{x}{5}dx
=\frac{1}{5!}\int_{0}^{4}x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)dx=0$$
- 3,112
3
votes
Is Beta function essential to evaluating the integral $\int_0^{\infty} \frac{d x}{\left(1+x^4\right)^n}$?
We are glad to have 3 nice and inspiring alternative solutions. I now want to give one more by introducing a reduction formula :
$$
I_{n+1}=\frac{4 n-1}{4 n} I_n \text { for any } n\in \mathbb{N}.
$$
...
- 18k
2
votes
Accepted
Values for which $\frac{K(k')}{K(k)}=a+i$ where $a$ is an algebraic number.
The ideas in your question related to complex values of $K'/K$ have been studied in past. In particular the theory of complex multiplication says that if $\tau$ is an algebraic number of degree $2$ ...
- 85.8k
1
vote
Mistake with Integration with Beta, Gamma, Digamma Fuctions
One-line solution using steps in More (Almost) Impossible Integrals, Sums, and Series (2023), page 193, the sequel of (Almost) Impossible Integrals, Sums, and Series (2019), for a similar integral. ...
- 5,169
1
vote
Calculation of a derivative of a function related to the Euler Gamma function
Too long for a comment. I am grateful to Inbo Gottlieb-Fenves for the nice answer. I am now realizing that I should have said that
$
F(x)=\frac2π\int_0^{π/2}(\cos \theta)^x d\theta,
$
so that
$$
\frac ...
- 699
1
vote
Calculation of a derivative of a function related to the Euler Gamma function
Set $f(x) = \Gamma(\frac{1+x}{2})$, $g(x) = \sqrt{\pi}\Gamma(1+\frac{x}{2})$. We compute the derivatives of $f$ and $g$ to get
$$f'(x) = \frac{1}{2}\int_0^\infty t^{\frac{x-1}{2}}e^{-t}\ln(t)\, dt \...
1
vote
Accepted
Question about derivative of the falling factorial
You have $f(n, n) = \Gamma(n + 1)$ but this by no means implies that $\frac d{dx}f(x, n)\vert_{x = n} = \Gamma'(n + 1)$.
Even if you can extend the arguments of $f$ to a differentiable function $f(x, ...
- 22k
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