The Gamma Function $y = \Gamma(x)$

The gamma function $\Gamma(x)$ extends the factorial $n!$ from the integers to the real (and even complex) numbers. For $x > 0$ it is defined by the integral

Relation to the factorial Integration by parts yields the recurrence $\Gamma(x+1) = x\,\Gamma(x)$, and for a positive integer $n$ one has $\Gamma(n) = (n-1)!$. Indeed $\Gamma(1) = 1$, $\Gamma(2) = 1$, $\Gamma(3) = 2$, and $\Gamma(4) = 6$ reproduce the factorials.

Domain The recurrence extends the function to $x \le 0$, but at $x = 0, -1, -2, \dots$ its value blows up. The domain is therefore all real numbers except these points.

Asymptotes and limits There is a vertical asymptote at each of $x = 0, -1, -2, \dots$. As $x \to +\infty$ the function grows explosively like the factorial, and as $x \to 0^+$ it diverges to $+\infty$. On the negative side it forms narrow spikes shooting to $\pm\infty$, alternating sign between consecutive integers.

Monotonicity and extremum For $x > 0$ the gamma function is convex (in fact logarithmically convex) and attains a minimum $\approx 0.8856$ at $x \approx 1.4616$. It decreases to the left of that point and increases to the right, and the valley lies between the two points $\Gamma(1) = \Gamma(2) = 1$.

Special values and identities At half-integers it takes the elegant value $\Gamma\!\left(\dfrac{1}{2}\right) = \sqrt{\pi}$, tied to the Gaussian integral. The reflection formula $\Gamma(x)\,\Gamma(1-x) = \dfrac{\pi}{\sin(\pi x)}$ holds, and substituting $x = \dfrac{1}{2}$ recovers that value.

Applications and history Euler introduced it in the 18th century as an interpolation of the factorial, and the notation $\Gamma$ is due to Legendre. It plays a fundamental role in probability (the gamma and beta distributions), complex analysis, combinatorics, and physics, making it one of the more advanced yet essential special functions.