The gamma and beta function

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The gamma and beta function

Γ ⁡ ( z ) := ∫ 0 ∞ t z - 1 ⁢ e - t ⁢ 𝑑 t assign Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 𝑡 differential-d 𝑡 {\displaystyle{\displaystyle{\displaystyle{}\Gamma\left(z\right):=\int_{0}^{% \infty}t^{z-1}{\mathrm{e}^{-t}}\,dt}}} {\displaystyle \index{Gamma function} \EulerGamma@{z}:=\int_0^{\infty}t^{z-1}\expe^{-t}\,dt }

Constraint(s): ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle{\displaystyle\Re{z}>0}}}


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 ) = π sin ⁡ ( π ⁢ z ) Euler-Gamma 𝑧 Euler-Gamma 1 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z% \right)=\frac{\pi}{\sin\left(\pi z\right)}}}} {\displaystyle \EulerGamma@{z}\EulerGamma@{1-z}=\frac{\cpi}{\sin@{\cpi z}} }

Constraint(s): z ∉ ℤ 𝑧 ℤ {\displaystyle{\displaystyle{\displaystyle z\notin\mathbb{Z}}}}


∫ - ∞ ∞ e - x 2 ⁢ 𝑑 x = 2 ⁢ ∫ 0 ∞ e - x 2 ⁢ 𝑑 x = ∫ 0 ∞ t - 1 / 2 ⁢ e - t ⁢ 𝑑 t = Γ ⁡ ( 1 / 2 ) = π superscript subscript superscript 𝑥 2 differential-d 𝑥 2 superscript subscript 0 superscript 𝑥 2 differential-d 𝑥 superscript subscript 0 superscript 𝑡 1 2 𝑡 differential-d 𝑡 Euler-Gamma 1 2 {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{e}^{% -x^{2}}}\,dx=2\int_{0}^{\infty}{\mathrm{e}^{-x^{2}}}\,dx=\int_{0}^{\infty}t^{-% 1/2}{\mathrm{e}^{-t}}\,dt=\Gamma\left(1/2\right)=\sqrt{\pi}}}} {\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} }
∫ - ∞ ∞ e - α 2 ⁢ x 2 - 2 ⁢ β ⁢ x ⁢ 𝑑 x = π α 2 ⁢ e β 2 / α 2   α , β ∈ ℝ formulae-sequence superscript subscript superscript 𝛼 2 superscript 𝑥 2 2 𝛽 𝑥 differential-d 𝑥 superscript 𝛼 2 superscript 𝛽 2 superscript 𝛼 2 𝛼 𝛽 ℝ {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{e}^{% -\alpha^{2}x^{2}-2\beta x}}\,dx=\sqrt{\frac{\pi}{\alpha^{2}}}\,{\mathrm{e}^{% \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} }

Constraint(s): α ≠ 0 𝛼 0 {\displaystyle{\displaystyle{\displaystyle\alpha\neq 0}}}


Γ ⁡ ( z ) ⁢ Γ ⁡ ( z + 1 / 2 ) = 2 1 - 2 ⁢ z ⁢ π ⁢ Γ ⁡ ( 2 ⁢ z ) , z ∈ ℂ formulae-sequence Euler-Gamma 𝑧 Euler-Gamma 𝑧 1 2 superscript 2 1 2 𝑧 Euler-Gamma 2 𝑧 𝑧 ℂ {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(z+1/% 2\right)=2^{1-2z}\sqrt{\pi}\,\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} }

Constraint(s): 2 ⁢ z ≠ 0 , - 1 , - 2 , … 2 𝑧 0 1 2 … {\displaystyle{\displaystyle{\displaystyle 2z\neq 0,-1,-2,\ldots}}}


Γ ⁡ ( z ) ∼ 2 ⁢ π ⁢ z z - 1 / 2 ⁢ e - z similar-to Euler-Gamma 𝑧 2 superscript 𝑧 𝑧 1 2 𝑧 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\sim\sqrt{2\pi}% \,z^{z-1/2}{\mathrm{e}^{-z}}}}} {\displaystyle \EulerGamma@{z}\sim\sqrt{2\cpi}\,z^{z-1/2}\expe^{-z} }

Constraint(s): ℜ ⁡ z → ∞ → 𝑧 {\displaystyle{\displaystyle{\displaystyle\Re{z}\rightarrow\infty}}}


Γ ⁡ ( x + i ⁢ y ) ∼ 2 ⁢ π ⁢ | y | x - 1 / 2 ⁢ e - | y | ⁢ π / 2 similar-to Euler-Gamma 𝑥 imaginary-unit 𝑦 2 superscript 𝑦 𝑥 1 2 𝑦 2 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(x+\mathrm{i}y\right)\sim% \sqrt{2\pi}\,|y|^{x-1/2}{\mathrm{e}^{-|y|\pi/2}}}}} {\displaystyle \EulerGamma@{x+\iunit y}\sim\sqrt{2\cpi}\,|y|^{x-1/2}\expe^{-|y|\cpi/2} }

Constraint(s): | y | → ∞ → 𝑦 {\displaystyle{\displaystyle{\displaystyle|y|\rightarrow\infty}}}


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} }

Constraint(s): | z | → ∞ → 𝑧 {\displaystyle{\displaystyle{\displaystyle|z|\rightarrow\infty}}}


B ⁡ ( x , y ) := ∫ 0 1 t x - 1 ⁢ ( 1 - t ) y - 1 ⁢ 𝑑 t , ℜ ⁡ 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 𝑥 0 {\displaystyle{\displaystyle{\displaystyle{}\mathrm{B}\left(x,y\right):=\int_{% 0}^{1}t^{x-1}(1-t)^{y-1}\,dt,\quad\Re{x}>0}}} {\displaystyle \index{Beta function} \EulerBeta@{x}{y}:=\int_0^1t^{x-1}(1-t)^{y-1}\,dt,\quad\realpart{x}>0 }

Constraint(s): ℜ ⁡ y > 0 𝑦 0 {\displaystyle{\displaystyle{\displaystyle\Re{y}>0}}}


B ⁡ ( x , y ) = Γ ⁡ ( x ) ⁢ Γ ⁡ ( y ) Γ ⁡ ( x + y )   ℜ ⁡ x > 0 formulae-sequence Euler-Beta 𝑥 𝑦 Euler-Gamma 𝑥 Euler-Gamma 𝑦 Euler-Gamma 𝑥 𝑦 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\Re{x}>0% }}} {\displaystyle \EulerBeta@{x}{y}=\frac{\EulerGamma@{x}\EulerGamma@{y}}{\EulerGamma@{x+y}} \quad\realpart{x}>0 }

Constraint(s): ℜ ⁡ y > 0 𝑦 0 {\displaystyle{\displaystyle{\displaystyle\Re{y}>0}}}


1 2 ⁢ π ⁢ ∫ - ∞ ∞ d ⁢ t ( r + i ⁢ t ) ρ ⁢ ( s - i ⁢ t ) σ = ( r + s ) 1 - ρ - σ ⁢ Γ ⁡ ( ρ + σ - 1 ) Γ ⁡ ( ρ ) ⁢ Γ ⁡ ( σ ) 1 2 superscript subscript 𝑑 𝑡 superscript 𝑟 imaginary-unit 𝑡 𝜌 superscript 𝑠 imaginary-unit 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\pi}\int_{-\infty}^{\infty% }\frac{dt}{(r+\mathrm{i}t)^{\rho}(s-\mathrm{i}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}} }

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