How do you simplify #(1.5)!#?
1 Answer
Mar 4, 2016
Explanation:
The factorial of a fractional number is defined by the gamma function as follows
#n! =n\times (n-1)!#
#Gamma (n)=(n-1)!#
#n! =n*Gamma(n)#
and
#Gamma (1/2)=sqrtpi#
Hence
#(1.5)! =(3/2)! =(3/2)* (1/2)! =(3/2)* (1/2) *Gamma(1/2)=3/4*sqrtpi#
Another way is to use Gauss's duplication formula which is defined as
#(n+1/2)! =\frac{\sqrt{\pi}(2n+2)!}{4^{n+1}(n+1)!}#
This allows you to express factorials of fractional number in terms of factorials of integers.
Now for
#(1+1/2)! =[sqrtpi*4!]/[4^2*2!]=[sqrtpi*24]/[32]=barul|(3sqrtpi)/4|#