3.4.1 Gamma function

The Gamma function, $\mathrm{\Gamma }(r)$, was covered in the Year 1 probability module and is defined as:

The two key properties are

1. $\mathrm{\Gamma }(1)={\int }_0^{\mathrm{\infty }}\exp (-t)dt={\left[-\exp (-t)\right]}_0^{\mathrm{\infty }}=1$,

2. $\mathrm{\Gamma }(r+1)=r\mathrm{\Gamma }(r)$ for $r>0$.

The second result was proved in your Year 1 probability module using integration by parts; the proof is reproduced in Appendix B. The appendix also details some further properties of $\mathrm{\Gamma }(r)$. The two results imply that when $r$ is an integer, $\mathrm{\Gamma }(r)=(r-1)!$. Later in this chapter we will also require: $\mathrm{\Gamma }(1/2)=\sqrt{\pi }$, which is also proved in Appendix B.