The general formula for expressing $\Gamma$(n+$\frac{1}{2}$) for n $\in$ $\mathbb{Z}$ in factorials is
$\Gamma$(n+$\frac{1}{2}$) = $\frac{(2n-1)!!}{2^n}\sqrt{\pi}$, where n is a non-negative integer and !! denotes the double factorial function.
The formula for expressing $\Gamma$(n+$\frac{1}{2}$) in factorials is derived using the properties of the gamma function and the definition of the double factorial function. It involves substituting $\frac{1}{2}$ for the variable x in the standard formula for $\Gamma$(x), and then simplifying the resulting expression.
The expression of $\Gamma$(n+$\frac{1}{2}$) in factorials is significant because it allows for the evaluation of the gamma function at half-integer values, which are commonly encountered in mathematical and scientific problems. It also provides a simplified form for calculating integrals involving the gamma function.
Examples of expressing $\Gamma$(n+$\frac{1}{2}$) in factorials include:
- $\Gamma$(1+$\frac{1}{2}$) = $\frac{1}{2}\sqrt{\pi}$
- $\Gamma$(2+$\frac{1}{2}$) = $\frac{3}{4}\sqrt{\pi}$
- $\Gamma$(3+$\frac{1}{2}$) = $\frac{15}{8}\sqrt{\pi}$
- $\Gamma$(4+$\frac{1}{2}$) = $\frac{105}{16}\sqrt{\pi}$
and so on.
The formula for expressing $\Gamma$(n+$\frac{1}{2}$) in factorials is used in various mathematical and scientific fields, such as probability theory, statistics, and physics. It allows for the calculation of various integrals involving half-integer values, which arise in many real-life problems. Additionally, it is also used in the development of numerical algorithms for computing the gamma function.