quality:corruption
General

Besides the fundamental property discussed above: Γ (z + 1) = z Γ (z) {\displaystyle \Gamma (z+1)=z\ \Gamma (z)} other important functional equations for the gamma function are Euler's reflection formula Γ (1 − z) Γ (z) = π sin π z, z ∉ Z {\displaystyle \Gamma (1-z)\Gamma (z)={\frac {\pi }{\sin \pi z}},\qquad z\not \in \mathbb {Z} } which implies Γ (z − n) = (− 1) n − 1 Γ (− z) Γ (1 + z) Γ (n + 1 − z), n ∈ Z {\displaystyle \Gamma (z-n)=(-1)^{n-1}\;{\frac {\Gamma (-z)\Gamma (1+z)}{\Gamma (n+1-z)}},\qquad n\in \mathbb {Z} } and the Legendre duplication formula Γ (z) Γ (z + 1 2) = 2 1 − 2 z π Γ (2 z). {\displaystyle \Gamma (z)\Gamma \left(z+{\tfrac {1}{2}}\right)=2^{1-2z}\;{\sqrt {\pi }}\;\Gamma (2z).}

The duplication formula is a special case of the multiplication theorem (see Eq. 5.5.6): ∏ k = 0 m − 1 Γ (z + k m) = (2 π) m − 1 2 m 1 2 − m z Γ (m z). {\displaystyle \prod _{k=0}^{m-1}\Gamma \left(z+{\frac {k}{m}}\right)=(2\pi)^{\frac {m-1}{2}}\;m^{{\frac {1}{2}}-mz}\;\Gamma (mz).}

A simple but useful property, which can be seen from the limit definition, is: Γ (z) ¯ = Γ (z ¯) ⇒ Γ (z) Γ (z ¯) ∈ R. {\displaystyle {\overline {\Gamma (z)}}=\Gamma ({\overline {z}})\;\Rightarrow \;\Gamma (z)\Gamma ({\overline {z}})\in \mathbb {R}.}

In particular, with z = a + bi, this product is | Γ (a + b i) | 2 = | Γ (a) | 2 ∏ k = 0 ∞ 1 1 + b 2 (a + k) 2 {\displaystyle |\Gamma (a+bi)|^{2}=|\Gamma (a)|^{2}\prod _{k=0}^{\infty }{\frac {1}{1+{\frac {b^{2}}{(a+k)^{2}}}}}}

If the real part is an integer or a half-integer, this can be finitely expressed in closed form: | Γ (b i) | 2 = π b sinh π b | Γ (1 2 + b i) | 2 = π cosh π b | Γ (1 + b i) | 2 = π b sinh π b | Γ (1 + n + b i) | 2 = π b sinh π b ∏ k = 1 n (k 2 + b 2), n ∈ N | Γ (− n + b i) | 2 = π b sinh π b ∏ k = 1 n (k 2 + b 2) − 1, n ∈ N | Γ (1 2 ± n + b i) | 2 = π cosh π b ∏ k = 1 n ((k − 1 2) 2 + b 2) ± 1, n ∈ N {\displaystyle {\begin{aligned}|\Gamma (bi)|^{2}&={\frac {\pi }{b\sinh \pi b}}\\\left|\Gamma \left({\tfrac {1}{2}}+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\\\left|\Gamma \left(1+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\\\left|\Gamma \left(1+n+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right),\quad n\in \mathbb {N} \\\left|\Gamma \left(-n+bi\right)\right|^{2}&={\frac {\pi }{b\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right)^{-1},\quad n\in \mathbb {N} \\\left|\Gamma \left({\tfrac {1}{2}}\pm n+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\prod _{k=1}^{n}\left(\left(k-{\tfrac {1}{2}}\right)^{2}+b^{2}\right)^{\pm 1},\quad n\in \mathbb {N} \\&\end{aligned}}}

Perhaps the best-known value of the gamma function at a non-integer argument is Γ (1 2) = π, {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)={\sqrt {\pi }},} which can be found by setting z = 1 2 {\textstyle z={\frac {1}{2}}} in the reflection or duplication formulas, by using the relation to the beta function given below with z 1 = z 2 = 1 2 {\textstyle z_{1}=z_{2}={\frac {1}{2}}}, or simply by making the substitution u = z {\displaystyle u={\sqrt {z}}} in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of n {\displaystyle n} we have: Γ (1 2 + n) = (2 n) ! 4 n n ! π = (2 n − 1) ! ! 2 n π = (n − 1 2 n) n ! π Γ (1 2 − n) = (− 4) n n ! (2 n) ! π = (− 2) n (2 n − 1) ! ! π = π (− 1 / 2 n) n ! {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}n!{\sqrt {\pi }}\\\Gamma \left({\tfrac {1}{2}}-n\right)&={(-4)^{n}n! \over (2n)!}{\sqrt {\pi }}={\frac {(-2)^{n}}{(2n-1)!!}}{\sqrt {\pi }}={\frac {\sqrt {\pi }}{{\binom {-1/2}{n}}n!}}\end{aligned}}} where the double factorial (2 n − 1) ! ! = (2 n − 1) (2 n − 3) ⋯ (3) (1) {\displaystyle (2n-1)!!=(2n-1)(2n-3)\cdots (3)(1)}. See Particular values of the gamma function for calculated values.

It might be tempting to generalize the result that Γ (1 2) = π {\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}} by looking for a formula for other individual values Γ (r) {\displaystyle \Gamma (r)} where r {\displaystyle r} is rational, especially because according to Gauss's digamma theorem, it is possible to do so for the closely related digamma function at every rational value. However, these numbers Γ (r) {\displaystyle \Gamma (r)} are not known to be expressible by themselves in terms of elementary functions. It has been proved that Γ (n + r) {\displaystyle \Gamma (n+r)} is a transcendental number and algebraically independent of π {\displaystyle \pi } for any integer n {\displaystyle n} and each of the fractions r = 1 6, 1 4, 1 3, 2 3, 3 4, 5 6 {\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}}. In general, when computing values of the gamma function, we must settle for numerical approximations.

The derivatives of the gamma function are described in terms of the polygamma function, ψ (z): Γ ′ (z) = Γ (z) ψ (0) (z). {\displaystyle \Gamma '(z)=\Gamma (z)\psi ^{(0)}(z).} For a positive integer m the derivative of the gamma function can be calculated as follows:

Γ ′ (m + 1) = m ! (− γ + ∑ k = 1 m 1 k) = m ! (− γ + H (m)), {\displaystyle \Gamma '(m+1)=m!\left(-\gamma +\sum _{k=1}^{m}{\frac {1}{k}}\right)=m!\left(-\gamma +H(m)\right)\,,} where H(m) is the mth harmonic number and γ is the Euler–Mascheroni constant.

For ℜ (z) > 0 {\displaystyle \Re (z)>0} the n {\displaystyle n} th derivative of the gamma function is: d n d z n Γ (z) = ∫ 0 ∞ t z − 1 e − t (log t) n d t. {\displaystyle {\frac {d^{n}}{dz^{n}}}\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}(\log t)^{n}\,dt.} (This can be derived by differentiating the integral form of the gamma function with respect to z {\displaystyle z}, and using the technique of differentiation under the integral sign.)

Using the identity Γ (n) (1) = (− 1) n B n (γ, 1 ! ζ (2), …, (n − 1) ! ζ (n)) {\displaystyle \Gamma ^{(n)}(1)=(-1)^{n}B_{n}(\gamma,1!\zeta (2),\ldots,(n-1)!\zeta (n))} where ζ (z) {\displaystyle \zeta (z)} is the Riemann zeta function, and B n {\displaystyle B_{n}} is the n {\displaystyle n} -th Bell polynomial, we have in particular the Laurent series expansion of the gamma function Γ (z) = 1 z − γ + 1 2 (γ 2 + π 2 6) z − 1 6 (γ 3 + γ π 2 2 + 2 ζ (3)) z 2 + O (z 3). {\displaystyle \Gamma (z)={\frac {1}{z}}-\gamma +{\frac {1}{2}}\left(\gamma ^{2}+{\frac {\pi ^{2}}{6}}\right)z-{\frac {1}{6}}\left(\gamma ^{3}+{\frac {\gamma \pi ^{2}}{2}}+2\zeta (3)\right)z^{2}+O(z^{3}).}