How do you find ( -3/2)factorial?

1 Answer
George C.
Aug 8, 2016

#(-3/2)! = Gamma(-1/2) = -2sqrt(pi)#

Explanation:

Strictly speaking, factorial is only defined for non-negative integers, but its definition is extended to other values using the #Gamma# function. For positive numbers and Complex numbers with positive Real part, we have:

#Gamma(t) = int_(x=0)^oo x^(t-1) e^(-x) dx#

Which satisfies:

#n! = Gamma(n+1)color(white)(X)# for any non-negative integer #n#.

#Gamma(1/2) = sqrt(pi)/2#

In the case of #-3/2#, we can (sort of) write:

#(-3/2)! = Gamma(-3/2+1)#

#= (Gamma(-1/2))/(-1/2)#

#= (Gamma(1/2))/(1/2*(-1/2))#

#= (sqrt(pi)/2)/(-1/4)#

#= -4(sqrt(pi)/2)#

#= -2sqrt(pi)#

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