Improper integral in the context of Riemann integral

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ${\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}$ is said to converge absolutely if ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L}$ for some real number ${\displaystyle \textstyle L.}$ Similarly, an improper integral of a function, ${\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx,}$ is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ${\displaystyle \textstyle \int _{0}^{\infty }|f(x)|dx=L.}$ A convergent series that is not absolutely convergent is called conditionally convergent.

Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergent series.

In mathematics, the gamma function (represented by ${\displaystyle \Gamma }$, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function ${\displaystyle \Gamma (z)}$ is defined for all complex numbers ${\displaystyle z}$ except non-positive integers, and ${\displaystyle \Gamma (n)=(n-1)!}$ for every positive integer ${\displaystyle n}$. The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

$${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.}$$The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.