Why is the Gamma function defined the way it is?

Question

The Gamma function $\Gamma(n)$ is defined as $$\Gamma(n)=\int_{0}^{\infty}x^{n-1}e^{-x}dx$$ $$\Gamma(n)=(n-1)!$$ Why isn't it defined as $$f(n)=\int_{0}^{\infty}x^{n}e^{-x}dx$$ $$f(n)=n!$$ This seems a lot more intuitive and easier to apply, or is there something I'm missing in terms of uses for the function?

Answer 1

I prefer to write this as an answer instead of a comment because this paper answer exactly the question and have historical references and bibliography about, and this paper is not linked in other answers about this same topic:

Gronau - Why is the Gamma function so as it is?

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Summary: the second Euler integral (the extension of the factorial) was represented earlier by Euler (derived from an earlier representation of his first Euler integral) as

$$\Delta(a):=\int_0^1(-\ln x)^a\mathrm dx\tag{1}$$

because the notation of the factorial $n!$ was introduced a bit later (but in the time of Euler anyway) so instead he used $\Delta(a)=\Gamma(a+1)$. So the Euler natural notation was a direct extension of $n!$.

But Legendre derived the extension of the factorial function from one of the latest notations that Euler used for his first integral:

$$\int_0^1\frac{x^{p-1}}{\sqrt[n]{(1-x)^{n-q}}}\mathrm dx\tag{2}$$

where if we set $n=1$ in $(2)$ then we have our actual definition for the beta function. From $(2)$ Legendre derived the actual notation of the gamma function

$$\Gamma(a):=\int_0^1\left(\ln \frac1x\right)^{a-1}\mathrm dx$$

(but the linked paper doesnt show this derivation.)

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Conclusions: the conclusions of the paper seems to be that the actual gamma notation is due to the Legendre derivation (searching an extension of the factorial function) from the first Euler integral written as $(2)$.

Moreover: from $(2)$ the paper seems to say that Euler himself derived his first integral as

$$\int_0^1\left(\ln\frac1x\right)^{a-1}\mathrm dx\tag{3}$$ in some time after he write $(1)$. So in the end it seems that Euler created both notations, the earlier $(1)$ was explicit, the later $(3)$ seems to be implicit as the paper suggest.