Consider the infinite product $H(z)=z\prod _{n=1}^{\mathrm{\infty }}(1+\frac{z}{n})e^{-z/n}$. One can show that this infinite product converges. Furthermore, it satisfies an identity:

where $\gamma$, also called Euler's constant, is the limit of the sequence $\sum _{n=1}^N\frac{1}{n}-\ln (N)$.

Furthermore, we may define $\mathrm{\Gamma }(z)=\frac{e^{-\gamma z}}{H(z)}$. Then $\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)$, and we see that this is in fact the famous Gamma function. Using this decomposition, one can also check that:

with the help of the previous formula for $\sin (\pi z)$.