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I asked this question on MSE here

$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ x>0. $$

Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties

  1. $f(x+1)=xf(x)$
  2. $f(1)=1$
  3. $f(x)$ is a continuous
  4. $f(x)$ is log convex

But before Bohr and Mollerup why did Euler or Bernoulli chose this function ? there are infinite many positive continuous function $f$ st $f(x+1)=xf(x)$, was there any alternatives functions for the gamma function as a generalisation for the factorials that was used before and now we stopped? If there is one then why do we stop using it? If there isn't one then how did Euler and other found the "Only" "natural" generalisation for the factorials centuries before Bohr and Mollerup as somehow they knew that this solution is unique and why $\Gamma$ stands out from other options.

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    $\begingroup$ Euler did not "choose" because he had nothing to choose from, see Davis. In the 18th century, "function" meant analytic formula, so he played around with infinite products and found one that interpolated the factorial. Alternatives have more complex analytic expressions and did not come up. Appearance of $\pi$ for half-integers suggested integrals similar to those for the circle area, which he then transformed into a form of the $\Gamma$ integral. Not the OP one, that one was derived from his later by Legendre. $\endgroup$
    – Conifold
    Mar 20 at 23:57
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    $\begingroup$ By mid-19th century one could trivially write down alternatives by adding to $\Gamma$ a function that vanishes on all integers (Weierstrass factorization theorem constructs such analytic functions as modified infinite products), but there was little point to the exercise. The result is more complicated than $\Gamma$, and $\Gamma$ had many connections to other functions of interest, aside from interpolating the factorial, to be singled out anyway. $\endgroup$
    – Conifold
    Mar 21 at 0:07
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    $\begingroup$ @pie Not my area of expertise, but you may enjoy reading the following account: Peter Luschny, "Is the Gamma function mis-defined? Or: Hadamard versus Euler - Who found the better Gamma function?" $\endgroup$
    – njuffa
    Mar 21 at 2:48
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    $\begingroup$ Leonhard Euler's integral: a historical profile of the gamma function by Philip J. Davis (1959) is freely available and might be useful for what you're interested in. Incidentally, regarding the comment about the gamma function being a transcendentally transcendental function (bottom half of p. 864), see this MSE answser. $\endgroup$ Mar 21 at 11:16
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    $\begingroup$ I just noticed (over an hour later) that @Conifold had already cited the Davis paper. I did read the previous comments before writing mine, but I had missed the "see Davis" part at the end of his first comment's first sentence. $\endgroup$ Mar 21 at 13:07

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