Gamma Poles
y = gamma(x)
Open in the app The dials and keys named below are the app's.
What it draws
The gamma function, which continues the factorial to every number rather than the whole ones. On the positive side it passes through the factorials exactly: gamma(1) = 1, gamma(3) = 2, gamma(4) = 6, gamma(5) = 24, each one (x − 1)!. Between them it rises smoothly, and at the halves it gives gamma(0.5) = 1.772, which is √π.
The whole curve rests on one relation, gamma(x + 1) = x · gamma(x). That is the recurrence the factorial obeys, and it is what makes the continuation the natural one rather than merely a curve through the right points.
The poles
Read that relation backwards, as gamma(x) = gamma(x + 1)/x, and the left-hand side follows. Approaching zero it divides a value near 1 by something vanishing, so the curve runs off. Step left again and the same division repeats against the spike already there, with the sign flipped each time. The result is a pole at every whole number from zero down, alternating in direction, and the plot shows four of them inside its span.
The gaps between them stay finite. At x = −0.5 the value is −3.545, and at x = −1.5 it is 2.363, both perfectly ordinary numbers sitting between two infinities.
Try
- Type integral(u^(x − 1) · exp(−u), u, 0, inf), which is Euler's integral and the app's other gamma preset. It agrees on the right. To the left of x = 0 the integral diverges, so that half pins to the top of the plot rather than tracing anything.
- Check the recurrence with gamma(x + 1)/x. The same curve, poles and all.
- Type gamma(x) · gamma(1 − x). The reflection formula says this is π/sin(πx), so the product is a chain of spikes with no smooth stretch anywhere.
- Raise Amplitude to 1. The poles leave the top immediately and the gentle part between them is what remains readable.