Γ ( z ) := ∫ 0 ∞ t z - 1 \expe - t 𝑑 t assign Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 superscript \expe 𝑡 differential-d 𝑡 {\displaystyle{\displaystyle{\displaystyle{}\Gamma\left(z\right):=\int_{0}^{% \infty}t^{z-1}\expe^{-t}\,dt}}} {\displaystyle \index{Gamma function} \EulerGamma@{z}:=\int_0^{\infty}t^{z-1}\expe^{-t}\,dt }
Γ ( z + 1 ) = z Γ ( z ) with Γ ( 1 ) = 1 formulae-sequence Euler-Gamma 𝑧 1 𝑧 Euler-Gamma 𝑧 with Euler-Gamma 1 1 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z+1\right)=z\Gamma\left(% z\right)\quad\textrm{with}\quad\Gamma\left(1\right)=1}}} {\displaystyle \EulerGamma@{z+1}=z\EulerGamma@{z}\quad\textrm{with}\quad\EulerGamma@{1}=1 } Γ ( z ) Γ ( 1 - z ) = \cpi sin ( \cpi z ) Euler-Gamma 𝑧 Euler-Gamma 1 𝑧 \cpi \cpi 𝑧 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z% \right)=\frac{\cpi}{\sin\!\left(\cpi z\right)}}}} {\displaystyle \EulerGamma@{z}\EulerGamma@{1-z}=\frac{\cpi}{\sin@{\cpi z}} }
∫ - ∞ ∞ \expe - x 2 𝑑 x = 2 ∫ 0 ∞ \expe - x 2 𝑑 x = ∫ 0 ∞ t - 1 / 2 \expe - t 𝑑 t = Γ ( 1 / 2 ) = \cpi superscript subscript superscript \expe superscript 𝑥 2 differential-d 𝑥 2 superscript subscript 0 superscript \expe superscript 𝑥 2 differential-d 𝑥 superscript subscript 0 superscript 𝑡 1 2 superscript \expe 𝑡 differential-d 𝑡 Euler-Gamma 1 2 \cpi {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\expe^{-x^{2}% }\,dx=2\int_{0}^{\infty}\expe^{-x^{2}}\,dx=\int_{0}^{\infty}t^{-1/2}\expe^{-t}% \,dt=\Gamma\left(1/2\right)=\sqrt{\cpi}}}} {\displaystyle \int_{-\infty}^{\infty}\expe^{-x^2}\,dx=2\int_0^{\infty}\expe^{-x^2}\,dx =\int_0^{\infty}t^{-1/2}\expe^{-t}\,dt=\EulerGamma@{1/2}=\sqrt{\cpi} } ∫ - ∞ ∞ \expe - α 2 x 2 - 2 β x 𝑑 x = \cpi α 2 \expe β 2 / α 2 α , β ∈ ℝ formulae-sequence superscript subscript superscript \expe superscript 𝛼 2 superscript 𝑥 2 2 𝛽 𝑥 differential-d 𝑥 \cpi superscript 𝛼 2 superscript \expe superscript 𝛽 2 superscript 𝛼 2 𝛼 𝛽 ℝ {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\expe^{-% \alpha^{2}x^{2}-2\beta x}\,dx=\sqrt{\frac{\cpi}{\alpha^{2}}}\,\expe^{\beta^{2}% /\alpha^{2}}\quad\alpha,\beta\in\mathbb{R}}}} {\displaystyle \int_{-\infty}^{\infty}\expe^{-\alpha^2x^2-2\beta x}\,dx =\sqrt{\frac{\cpi}{\alpha^2}}\,\expe^{\beta^2/\alpha^2} \quad\alpha,\beta\in\mathbb{R} }
Γ ( z ) Γ ( z + 1 / 2 ) = 2 1 - 2 z \cpi Γ ( 2 z ) , z ∈ ℂ formulae-sequence Euler-Gamma 𝑧 Euler-Gamma 𝑧 1 2 superscript 2 1 2 𝑧 \cpi Euler-Gamma 2 𝑧 𝑧 ℂ {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(z+1/% 2\right)=2^{1-2z}\sqrt{\cpi}\,\Gamma\left(2z\right),\quad z\in\mathbb{C}}}} {\displaystyle \EulerGamma@{z}\EulerGamma@{z+1/2}=2^{1-2z}\sqrt{\cpi}\,\EulerGamma@{2z},\quad z\in\mathbb{C} }
Γ ( z ) ∼ 2 \cpi z z - 1 / 2 \expe - z similar-to Euler-Gamma 𝑧 2 \cpi superscript 𝑧 𝑧 1 2 superscript \expe 𝑧 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\sim\sqrt{2\cpi}% \,z^{z-1/2}\expe^{-z}}}} {\displaystyle \EulerGamma@{z}\sim\sqrt{2\cpi}\,z^{z-1/2}\expe^{-z} }
Γ ( x + \iunit y ) ∼ 2 \cpi | y | x - 1 / 2 \expe - | y | \cpi / 2 similar-to Euler-Gamma 𝑥 \iunit 𝑦 2 \cpi superscript 𝑦 𝑥 1 2 superscript \expe 𝑦 \cpi 2 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(x+\iunit y\right)\sim% \sqrt{2\cpi}\,|y|^{x-1/2}\expe^{-|y|\cpi/2}}}} {\displaystyle \EulerGamma@{x+\iunit y}\sim\sqrt{2\cpi}\,|y|^{x-1/2}\expe^{-|y|\cpi/2} }
Γ ( z + a ) Γ ( z + b ) ∼ z a - b , a , b ∈ ℂ formulae-sequence similar-to Euler-Gamma 𝑧 𝑎 Euler-Gamma 𝑧 𝑏 superscript 𝑧 𝑎 𝑏 𝑎 𝑏 ℂ {\displaystyle{\displaystyle{\displaystyle\frac{\Gamma\left(z+a\right)}{\Gamma% \left(z+b\right)}\sim z^{a-b},\quad a,b\in\mathbb{C}}}} {\displaystyle \frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\sim z^{a-b},\quad a,b\in\mathbb{C} }
B ( x , y ) := ∫ 0 1 t x - 1 ( 1 - t ) y - 1 𝑑 t , \realpart x > 0 formulae-sequence assign Euler-Beta 𝑥 𝑦 superscript subscript 0 1 superscript 𝑡 𝑥 1 superscript 1 𝑡 𝑦 1 differential-d 𝑡 \realpart 𝑥 0 {\displaystyle{\displaystyle{\displaystyle{}\mathrm{B}\left(x,y\right):=\int_{% 0}^{1}t^{x-1}(1-t)^{y-1}\,dt,\quad\realpart{x}>0}}} {\displaystyle \index{Beta function} \EulerBeta@{x}{y}:=\int_0^1t^{x-1}(1-t)^{y-1}\,dt,\quad\realpart{x}>0 }
B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) \realpart x > 0 formulae-sequence Euler-Beta 𝑥 𝑦 Euler-Gamma 𝑥 Euler-Gamma 𝑦 Euler-Gamma 𝑥 𝑦 \realpart 𝑥 0 {\displaystyle{\displaystyle{\displaystyle\mathrm{B}\left(x,y\right)=\frac{% \Gamma\left(x\right)\Gamma\left(y\right)}{\Gamma\left(x+y\right)}\quad% \realpart{x}>0}}} {\displaystyle \EulerBeta@{x}{y}=\frac{\EulerGamma@{x}\EulerGamma@{y}}{\EulerGamma@{x+y}} \quad\realpart{x}>0 }
1 2 \cpi ∫ - ∞ ∞ d t ( r + \iunit t ) ρ ( s - \iunit t ) σ = ( r + s ) 1 - ρ - σ Γ ( ρ + σ - 1 ) Γ ( ρ ) Γ ( σ ) 1 2 \cpi superscript subscript 𝑑 𝑡 superscript 𝑟 \iunit 𝑡 𝜌 superscript 𝑠 \iunit 𝑡 𝜎 superscript 𝑟 𝑠 1 𝜌 𝜎 Euler-Gamma 𝜌 𝜎 1 Euler-Gamma 𝜌 Euler-Gamma 𝜎 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\cpi}\int_{-\infty}^{% \infty}\frac{dt}{(r+\iunit t)^{\rho}(s-\iunit t)^{\sigma}}=\frac{(r+s)^{1-\rho% -\sigma}\Gamma\left(\rho+\sigma-1\right)}{\Gamma\left(\rho\right)\Gamma\left(% \sigma\right)}}}} {\displaystyle \frac{1}{2\cpi}\int_{-\infty}^{\infty}\frac{dt}{(r+\iunit t)^{\rho}(s-\iunit t)^{\sigma}} =\frac{(r+s)^{1-\rho-\sigma}\EulerGamma@{\rho+\sigma-1}}{\EulerGamma@{\rho}\EulerGamma@{\sigma}} }
