Incomplete gamma function

Remarks

For $s > 0$ and $x > 0$, $\gamma(s, x) + \Gamma(s, x) = \Gamma(s)$, the gamma function.

Example: for $s = 1$, $\gamma(1, x) = \int_0^x e^{-t}\, dt = 1 - e^{-x}$ and $\Gamma(1, x) = e^{-x}$, and their sum is $1 = \Gamma(1)$. Integration by parts gives $\gamma(s+1, x) = s\,\gamma(s, x) - x^s e^{-x}$.

Non-example of a valid argument: for $s = 0$ the lower integral $\int_0^x t^{-1} e^{-t}\, dt$ diverges at $0$, so $\gamma(0, x)$ is not defined. The upper function is still defined: $\Gamma(0, x) = E_1(x)$ is the exponential integral.

The quotient $P(s, x) = \gamma(s, x)/\Gamma(s)$ is the regularised lower incomplete gamma function, the distribution function of the gamma distribution with shape $s$ and scale $1$. Both functions extend to complex arguments: $\gamma(s, x)$ for $\operatorname{Re} s > 0$, and $\Gamma(s, x)$ by analytic continuation.