Alternative definition of Gamma function?

Question

The Gamma function is defined in terms of an integral as

The notation $Γ(t)$ is due to Legendre. If the real part of the complex number $t$ is positive $(Re(t) > 0)$, then the integral $$ \Gamma(t) = \int_0^\infty x^t e^{-x}\,\frac{{\rm d}x}{x} $$ converges absolutely, and is known as the Euler integral of the second kind (the Euler integral of the first kind defines the Beta function).

Can it be equivalently defined in terms of a recursive relation as $$ \Gamma(t+1)=t \Gamma(t)$$ $$\Gamma(1)=1$$ with some non-redundant conditions?

Thanks!

Answer 1

Baby Rudin gives the following definition:

$\forall x\in(0,+\infty):\ \Gamma(x+1)=x\Gamma(x);$

$\forall n\in{\Bbb N}:\ \Gamma(n+1)=n!;$

$\log\Gamma$ is convex in $(0,+\infty)$.

Answer 2

The two definitions are equivalent if $t$ is a an integer. If $t$ is an integer less than equal to zero neither object is defined and if $t$ is an integer greater or equal than one then the two definitions are equivalent. In all other cases the standard definition is defined whereas yours is not defined.