The factorial n! coincides with the gamma function at positive integer values. So if one equates the factorial function on all positive reals with the gamma function, then one can say that
(1/2)! = sqrt(pi)/2
even though the factorial function was originally only defined for the natural numbers.
Added after 14 minutes:
Noooo I'am wrong. It's Gamma(n+1) = n! and one does not equate the factorial with the gamma function. Sorry.
Can't say that I see how to do the integral by just looking at it. But you can see how the integral comes
from a property of the Gamma function called the 'Euler reflection formula':
change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)
change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)