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Question: How do you find \(\left( { - \dfrac{3}{2}} \right)\) factorial?...

How do you find (−32)\left( { - \dfrac{3}{2}} \right) factorial?

Explanation

Solution

In this problem we have given some fractional terms with negative signs. And we asked to find the factorial of the given term. But the factorial of negative real numbers, are complex numbers. So to find the factorial of a given negative term we are going to use the gamma function. The main concept of this problem is a gamma function.

Formula used:
n!=Γ(n+1)n! = \Gamma \left( {n + 1} \right), where nn is any non - negative integer.

Complete Step by Step Solution:
We can define factorial as only for non – negative integers. But the definition of factorial is extended to other values using the Γ\Gamma function.
For positive numbers and complex numbers with positive real part, we have,Γ(t)=∫x=0∞xt−1e−xdx\Gamma \left( t \right) = \int\limits_{x = 0}^\infty {{x^{t - 1}}{e^{ - x}}dx} which satisfies n!=Γ(n+1)n! = \Gamma \left( {n + 1} \right)andΓ(12)=π2\Gamma \left( {\dfrac{1}{2}} \right) = \dfrac{{\sqrt \pi }}{2}
In the case of−32 - \dfrac{3}{2}, we can apply factorial we get, (−32)!\left( { - \dfrac{3}{2}} \right)!applying formula,
(−32)!=Γ(−32+1)\left( { - \dfrac{3}{2}} \right)! = \Gamma \left( { - \dfrac{3}{2} + 1} \right), adding numbers inside the bracket of right hand side, and using gamma function, we get
(−32)!=Γ(−12)−12\left( {\dfrac{{ - 3}}{2}} \right)! = \dfrac{{\Gamma \left( { - \dfrac{1}{2}} \right)}}{{ - \dfrac{1}{2}}}
(−32)!=Γ(12)12(−12)\left( {\dfrac{{ - 3}}{2}} \right)! = \dfrac{{\Gamma \left( {\dfrac{1}{2}} \right)}}{{\dfrac{1}{2}\left( { - \dfrac{1}{2}} \right)}}
Applying the factΓ(12)=π2\Gamma \left( {\dfrac{1}{2}} \right) = \dfrac{{\sqrt \pi }}{2}in the numerator and multiplying the terms in the denominator, we get,
(−32)!=π2−14\left( { - \dfrac{3}{2}} \right)! = \dfrac{{\dfrac{{\sqrt \pi }}{2}}}{{ - \dfrac{1}{4}}}, denominator of a denominator is the numerator.
⇒(−32)!=−4(π2)\Rightarrow \left( { - \dfrac{3}{2}} \right)! = - 4\left( {\dfrac{{\sqrt \pi }}{2}} \right), cancelling the numbers in the right hand side, we get
⇒(−32)!=−2π\Rightarrow \left( { - \dfrac{3}{2}} \right)! = - 2\sqrt \pi

Therefore, (−32)!=−2π\left( { - \dfrac{3}{2}} \right)! = - 2\sqrt \pi . This is the required solution.

Note: Factorial is the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point.
While solving this problem there is a question that will arise, that is can gamma be negative. For, the gamma function is extended to all complex numbers, with its real part strictly greater than zero, except for at zero and negative integers. So gamma function is positive.