[VA] SRC #009 - Pi Day 2021 Special

[Albert Chan]

(03-16-2021 04:59 PM)robve Wrote:  Rewrite the equation to

$$ \int_0^x \left( \frac{\sin t}{t} \mathrm{e}^{t/\tan t} \right)^x\,dt - \frac{x^x}{\Gamma x} = 0 $$

After some hunting on the interval [3,π] we find the root x=π.

That makes this a remarkable equation, which I am not yet sure where it came from.

From identity: \(\displaystyle \int_0^{\pi} \left( \frac{\sin t}{t} \mathrm{e}^{t/\tan t} \right)^x\;dt = {\pi x^x \over x!}\qquad\) , x ≥ 0

AN INTEGRAL REPRESENTATION FOR THE LAMBERT W FUNCTION