Two Questions about Gamma Function Terminology

Question

Gamma function is also known as generalized factorial function .

1. Why does the term "generalized" have been used?

2. Why is the Gamma function called Euler's second integral?

Answer 1

The Factorial is defined only for integers: $$ n!=n(n-1)(n-2)\cdots3\cdot2\cdot1 $$ The Gamma function, $\Gamma(x)$, is defined for general complex arguments $$ \Gamma(z)=\int_0^\infty t^{z-1}\,e^{-t}\,\mathrm{d}t $$ where the integral converges if $\mathrm{Re}(z)\gt0$, but $\Gamma(x)$ can be analytically continued to the whole complex plane minus some isolated points using the reflection formula: $$ \Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)} $$

$\Gamma(x)$ is called the generalized factorial since $n!=\Gamma(n+1)$.

The Beta function and the Gamma function are called Euler's first and second integrals.