Greetings:
My Windows 8 stock calculator gives values for certain fractional arguments and even some negatives. E.g., 0.5! is given to be 0.866...
I did not know that! I have never tried "!" with anything but non-negative integers in a calculator. I notice that using negative integers gives "invalid input" but that (-0.5)!= 1.772.
Is there some mathematical merit to this? Or is it just a case where the algorithm is processing these arguments and has no input filter written into the calculator's program?
Thank you.
Rich B.
It looks to me like the "Gamma function", \(\displaystyle \Gamma(x)\). This is defined by
\(\displaystyle \Gamma(x)= \int_0^\infty t^{x- 1}e^{-t}dt\).
We can easily see that \(\displaystyle \Gamma(1)= \int_0^\infty t^0e^{-t}dt= \int_0^\infty e^{-t}dt= \left[-e^{-t}\right]_0^\infty= 1\).
Using integration by parts, with \(\displaystyle u= t^{x- 1}\) and \(\displaystyle dv= e^{-t}dt\), so that \(\displaystyle du= (x- 1)t^{x- 2} dt\) and \(\displaystyle v= -e^{-t}\) we have
\(\displaystyle \Gamma(x)=\)\(\displaystyle \left[t^{x- 1}(-e^{-t})\right]_0^\infty+ (x- 1)\int_0^\infty t^{x- 2}e^{-t}dt=\)\(\displaystyle (x- 1)\Gamma(x-1)\) so that, for x a positive integer, \(\displaystyle \Gamma(1)= 1\), \(\displaystyle \Gamma(2)= 1(\Gamma(1))= 1\), \(\displaystyle \Gamma(3)= 2\Gamma(2)= 2\), \(\displaystyle \Gamma(4)= 3\Gamma(3)= 3(2)= 6\), etc. In other words, for x a positive integer, \(\displaystyle \Gamma(x)= (x- 1)!\).
