Question

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

Answer 1

Hint: Given the expression in negative fraction form. We have to find the factorial of the negative fractional number. First, we will apply the gamma function to the expression which is extended to the range of a set of all real numbers. Then, determine the factorial of the number using the formula which satisfies the gamma function.

Formula used:
The gamma function to find the factorial of negative integers is given as:
\[\Gamma \left( t \right) = \int_0^\infty {{s^{t - 1}}{e^{ - t}}dt} \]
Which will satisfies the equation
\[s! = \Gamma \left( {s + 1} \right)\] where \[s\] is any non-negative integer.

Complete step-by-step answer:
We are given the number in fraction form. Apply the gamma function to find the factorial of the number.
\[\left( { - \dfrac{3}{2}} \right)! = \Gamma \left( { - \dfrac{3}{2} + 1} \right)\]
On simplifying the expression, we get:
$ \Rightarrow \left( { - \dfrac{3}{2}} \right)! = \Gamma \left( {\dfrac{{ - 3 + 2}}{2}} \right)$
$ \Rightarrow \left( { - \dfrac{3}{2}} \right)! = \Gamma \left( { - \dfrac{1}{2}} \right)$
Now, we will apply the formula $\Gamma \left( {n + 1} \right) = n \times \Gamma \left( n \right)$. Now, substitute $n = - \dfrac{1}{2}$ to compute the value of $\Gamma \left( { - \dfrac{1}{2}} \right)$
$ \Rightarrow \Gamma \left( { - \dfrac{1}{2}} \right) = \dfrac{{\Gamma \left( { - \dfrac{1}{2} + 1} \right)}}{{ - \dfrac{1}{2}}}$
On simplifying the expression, we get:
$ \Rightarrow \Gamma \left( { - \dfrac{1}{2}} \right) = \dfrac{{\Gamma \left( {\dfrac{{ - 1 + 2}}{2}} \right)}}{{ - \dfrac{1}{2}}}$
$ \Rightarrow \Gamma \left( { - \dfrac{1}{2}} \right) = \dfrac{{\Gamma \left( {\dfrac{1}{2}} \right)}}{{ - \dfrac{1}{2}}}$
$ \Rightarrow \Gamma \left( { - \dfrac{1}{2}} \right) = - 2 \times \Gamma \left( {\dfrac{1}{2}} \right)$
Now, we will substitute $\Gamma \left( {\dfrac{1}{2}} \right) = \dfrac{{\sqrt \pi }}{2}$ into the expression.
$ \Rightarrow \left( { - \dfrac{3}{2}} \right)! = - 2\sqrt \pi $

Final answer: Hence the factorial of \[\left( { - \dfrac{3}{2}} \right)\] is $ - 2\sqrt \pi $

Additional information: The gamma function is defined to find the factorial for the numbers in the form non-integer or fraction form. In this function, $\Gamma \left( x \right)$ is defined for all values of $x$ and using the recurrence formula for the factorial the gamma function is written as $\Gamma \left( {x + 1} \right) = x \times \Gamma \left( x \right)$ such that $x$ need not to be positive. Therefore, the function is used to calculate the factorial of a non-positive number.

Note:
 In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students mainly make mistakes while applying the formula for finding factorial of a fractional integer. In such types of questions, students mainly get confused to define the recurrence relationship to calculate the factorials.