fresh_42 said:You are required to show us, what you've tried. Are you allowed to use Euler's reflection formula? The proofs I've found all uses it.
The Double Factorial function, denoted by n!!, is a mathematical function that is defined as the product of all positive integers from 1 to n that have the same parity (i.e. either all odd or all even numbers). For example, 5!! = 1 * 3 * 5 = 15.
The Double Factorial function is related to the Gamma function through its recursive definition. The Gamma function is defined as Γ(n) = (n-1)!, while the Double Factorial function is defined as n!! = n * (n-2)!! for n > 1. This relationship is the key to proving the identity Γ(n+1/2) = √π * n!! / 2^n.
This identity is significant because it allows us to express the values of the Gamma function at half-integer arguments in terms of the square root of pi and the Double Factorial function. This makes it easier to evaluate and work with these types of values in mathematical and scientific calculations.
The proof for Γ(p+1/2) using Double Factorial and nΓ(n) has many practical applications, especially in fields such as statistics, physics, and engineering. It allows for a more efficient and accurate computation of values involving the Gamma function at half-integer arguments. These values often appear in various mathematical models and equations, making this proof a valuable tool for solving real-world problems.
Yes, there are other methods for proving Γ(p+1/2) = √π * n!! / 2^n. Some of these methods include using the Wallis product, the Euler reflection formula, and complex analysis techniques. However, the proof using Double Factorial and nΓ(n) is one of the simplest and most elegant approaches, making it a popular choice among mathematicians and scientists.