The gamma function is defined by

$$\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt.$$ What made the mathematicians define it like this instead of $$\Gamma(z)=\int_0^\infty t^{z}e^{-t}dt.$$ This would make the gamma function a "real" generalization of the factorial function $\Gamma(n)=n!$

  • 3
    $\begingroup$ See this math overflow question. $\endgroup$ Sep 27, 2019 at 23:22
  • $\begingroup$ @PeterForeman Thanks, the mathematical community should do something about this, it's irritating :) $\endgroup$ Sep 27, 2019 at 23:32
  • $\begingroup$ There are several questions about that on MSE as well, for example, here. $\endgroup$
    – A.Γ.
    Sep 27, 2019 at 23:38
  • $\begingroup$ You might want to listen to this one youtube.com/watch?v=CO6IZtCMpjk $\endgroup$
    – H. Gutsche
    Sep 28, 2019 at 0:42
  • $\begingroup$ The notation was introduced by Legendre and he used a different integral (but equivalent) to introduce it. You might want to have a look at A. M. Legendre, Exercises de Calcul Intégral (1811), p. 277. $\endgroup$
    – H. Gutsche
    Sep 28, 2019 at 1:39


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