The gamma function, denoted as Γ(n+1), is defined for any real number n greater than -1, not limited to integer values. The integral representation provided, ∫₀ⁿ xⁿ e^(-ax) dx = Γ(n+1) / a^(n+1), holds true for both integer and half-integer values of n. This flexibility allows for broader applications of the gamma function in various mathematical contexts.
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Denver Dang said:Homework Statement
I have a quick question about the gamma function.
According to my textbook it says:
\int_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}=\frac{\Gamma \left( n+1 \right)}{{{a}^{n+1}}}
where \Gamma is the gamma function. My question is, do n has to be an integer number, or can it also be a half-integer number for the gamma function to work ?
Thanks in advance.