---- > [!definition] Definition. ([[gamma function]]) > The **gamma function** $\Gamma: \mathbb{C} \cut \mathbb{Z}_{\leq 0} \to \mathbb{R}$ is given by $\Gamma(z):= \int _{0}^{\infty} t^{z-1} e^{-t} \, dt .$ > In the special case $z \in \mathbb{N}$, we have $\Gamma(z)=(z-1)!$. More generally, the [[gamma function]] 'interpolates' the factorial function and is thus viewed as an extension of it. > [!basicproperties] > - $\Gamma(z+1)=z \Gamma(z)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```