# Gamma Poles

Wave · y = f(x, t)

`y = gamma(x)`

[Open in the app](https://www.wavelace.com/app#p=121) · [This page](https://www.wavelace.com/presets/gamma-poles)

### 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.

### Read more

- [Gamma function](https://en.wikipedia.org/wiki/Gamma_function)
- [Particular values of the gamma function](https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function)
- [Reflection formula](https://en.wikipedia.org/wiki/Reflection_formula)
- [Zeros and poles](https://en.wikipedia.org/wiki/Zeros_and_poles)