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For proofs of the following results, see for example Chapters 12 and 13 of [2].

FormalPara Definition 4.1

For all \(z \in \mathbb {C}\) with \(\Re (z)>0\), we define

$$\displaystyle \begin{aligned} \varGamma(z) = \int_0^{\infty} e^{-t} t^{z-1} dt. \end{aligned}$$
FormalPara Theorem 4.2 (Weierstrass Product)

For all \(z\in {{\mathbb {C}}}\) ,

$$\displaystyle \begin{aligned} \frac{1}{\varGamma(z)} = z\, e^{\gamma z} \prod_{n=1}^{\infty} \left(1+\frac{z}{n}\right) e^{-z/n}, \end{aligned}$$

where γ = 0.5772156649… is the Euler constant.

Thus the function Γ is meromorphic on \({{\mathbb {C}}}\) , has no zero, and admits simple poles at z = 0, −1, −2, ….

FormalPara Proposition 4.3

For \(z\in {{\mathbb {C}}}\setminus (-{{\mathbb {N}}})\) , we have

$$\displaystyle \begin{aligned} \varGamma(z+1) = z \, \varGamma(z). \end{aligned}$$
FormalPara Corollary 4.4

For \(n\in {{\mathbb {N}}}\) , we have Γ(n + 1) = n!.

FormalPara Proposition 4.5

For \(z\in {{\mathbb {C}}}\setminus {{\mathbb {Z}}}\) ,

$$\displaystyle \begin{aligned} \varGamma(z)\, \varGamma(1-z) = \frac{\pi}{\sin \pi z}. \end{aligned}$$
FormalPara Corollary 4.6

We have \(\varGamma ({{\textstyle \frac {1}{2}}}) = \sqrt {\pi }\).

FormalPara Proposition 4.7 (Legendre Duplication Formula)

For \(z\in {{\mathbb {C}}}\setminus (-{{\textstyle \frac {1}{2}}}{{\mathbb {N}}})\) ,

$$\displaystyle \begin{aligned} \varGamma(z)\, \varGamma(z+{{\textstyle\frac{1}{2}}}) = {\pi}^{1/2}\, 2^{1-2z}\, \varGamma(2z). \end{aligned}$$
FormalPara Corollary 4.8

For \(z\in {{\mathbb {C}}}\setminus (2{{\mathbb {Z}}})\) , we have

$$\displaystyle \begin{aligned}\textstyle \varGamma(\frac{z}{2})/ \varGamma(\frac{1-z}{2}) = {\pi}^{-1/2}\, 2^{1-z}\, \cos{}(\frac{\pi z}{2})\, \varGamma(z). \end{aligned}$$
FormalPara Theorem 4.9 (Stirling Formula)

For fixed \(0\leqslant \vartheta < \pi \) and \(\left | arg(z)\right | < \vartheta \) , we have

$$\displaystyle \begin{aligned} \log \varGamma(z) = (z-{{\textstyle\frac{1}{2}}}) \log z -z -{{\textstyle\frac{1}{2}}} \log 2\pi + O(\left| z\right|{}^{-1}), \quad\left| z\right| \rightarrow +\infty. \end{aligned}$$
FormalPara Proposition 4.10

For fixed \(0\leqslant \vartheta < \pi \) and \(\left | arg(z)\right | < \vartheta \) , we have

$$\displaystyle \begin{aligned} \frac{\varGamma'(z)}{\varGamma(z)} = \log z + O(\left| z\right|{}^{-1}), \quad\left| z\right| \rightarrow +\infty. \end{aligned}$$
FormalPara Proof

See [1, footnote, p.57].