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Question: How do you simplify \(\left( 1.5 \right)!?\)...

How do you simplify (1.5)!?\left( 1.5 \right)!?

Explanation

Solution

A factorize is a number multiplied by all of the integers below it.
Example “Four factorial” =4!=4×3×2×1=24=4!=4\times 3\times 2\times 1=24
The factorial of a number ′n′'n' is denoted as ′n!′'n!' This is the product of all numbers from 11 to n.n.

Complete step by step solution:
As per the given problem,
You have to simplify (1.5)!\left( 1.5 \right)!
Here, the factorial of a fraction number is defined by the gamma function is defined by the gamma function as follows.
n!=n×(n−1)!n!=n\times \left( n-1 \right)!
Γ(n)=(n−1)!\Gamma \left( n \right)=\left( n-1 \right)!
n!=n.Γ(n)n!=n.\Gamma (n)
And
Γ(12)=π...(i)\Gamma \left( \dfrac{1}{2} \right)=\sqrt{\pi }...(i)
Hence, simplify you can solve (1.5)!\left( 1.5 \right)!
(1.5)!=(32)!=(32).(12)!\left( 1.5 \right)!=\left( \dfrac{3}{2} \right)!=\left( \dfrac{3}{2} \right).\left( \dfrac{1}{2} \right)!
=(32)(12)Γ(12)=\left( \dfrac{3}{2} \right)\left( \dfrac{1}{2} \right)\Gamma \left( \dfrac{1}{2} \right)
=34π=\dfrac{3}{4}\sqrt{\pi } as from equation (i)(i)
Γ(12)=π\Gamma \left( \dfrac{1}{2} \right)=\sqrt{\pi }

Note: You can either use gamma function identity either gauss’s duplication formula. For simplifying the given factorial. Note that for using the Gauss’s duplication formula put n=1n=1 and simplify.
You can also solve 1.5!1.5! by using gauss’s duplication formula which is defined as below.
(n+12)!=π(2n+2)!4n+1(n+1)!\left( n+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }\left( 2n+2 \right)!}{{{4}^{n+1}}\left( n+1 \right)!}
This allows you to express a fractional number in terms of factorials of integer.
Now, for n=1n=1 we get from the above formula.
(1+12)!=π.4!42.2!\left( 1+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }.4!}{{{4}^{2}}.2!}
=π.2416.2!=\dfrac{\sqrt{\pi }.24}{16.2!}
=π.2432=\dfrac{\sqrt{\pi }.24}{32}
Here, above 2424 and 3434 comes in a table of ′8′'8' so you can simplify it.
(1+12)!=π2432\left( 1+\dfrac{1}{2} \right)!=\dfrac{\sqrt{\pi }24}{32}
(1+12)!=3π4\left( 1+\dfrac{1}{2} \right)!=\dfrac{3\sqrt{\pi }}{4}
Hence,
The simplification of 1.5!1.5! is 3π4\dfrac{3\sqrt{\pi }}{4}