How do you simplify #(1.5)!#?

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Answer 1 · 2016-03-04T13:20:51

#(1.5)! =(3sqrtpi)/4#

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 #n=1# we get from the above formula

#(1+1/2)! =[sqrtpi*4!]/[4^2*2!]=[sqrtpi*24]/[32]=barul|(3sqrtpi)/4|#