How do you simplify $(1.5)!$ $(1.5)!$ ?

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Answer 1 · 2016-03-04T13:20:51

$(1.5)! = \frac{3 \sqrt{π}}{4}$ $(1.5)! =(3sqrtpi)/4$

The factorial of a fractional number is defined by the gamma function as follows

$n! = n \times (n - 1)!$ $n! =n\times (n-1)!$

$Gamma (n)=(n-1)!$

$n! =n*Gamma(n)$

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

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

How do you simplify (1.5)!(1.5)!(1.5)!?