expressed as H α ( 1 ) ( x ) = J − α ( x ) − e − α π i J α ( x ) i sin ⁡ α π , H α ( 2 ) ( x ) = J − α ( x ) − e α π i J α ( x ) − i sin ⁡ α π . {\displaystyle...

x ) = Pr ( X > x ) = { ( x m x ) α x ≥ x m , 1 x < x m , {\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha...

α ) ( x ) = 1 L 1 ( α ) ( x ) = − x + ( α + 1 ) L 2 ( α ) ( x ) = x 2 2 − ( α + 2 ) x + ( α + 1 ) ( α + 2 ) 2 L 3 ( α ) ( x ) = − x 3 6 + ( α + 3 ) x...

= α x t + ( 1 − α ) s t − 1 = α x t + α ( 1 − α ) x t − 1 + ( 1 − α ) 2 s t − 2 = α [ x t + ( 1 − α ) x t − 1 + ( 1 − α ) 2 x t − 2 + ( 1 − α ) 3 x t...

mean α > 0. By n-fold application of the above inequality, we obtain that x 1 α x 2 α ⋯ x n α ≤ e x 1 α − 1 e x 2 α − 1 ⋯ e x n α − 1 = exp ⁡ ( x 1 α − 1...

}}\;x>0} x − 1 x ≤ ln ⁡ x ≤ x − 1 for x > 0 {\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0} ln ⁡ ( 1 + x α ) ≤ α x for x ≥...

α = 1 + α x + 1 2 α ( α − 1 ) x 2 + 1 6 α ( α − 1 ) ( α − 2 ) x 3 + 1 24 α ( α − 1 ) ( α − 2 ) ( α − 3 ) x 4 + ⋯ {\displaystyle {\begin{aligned}f(x)&=\sum...

B ( α , β ) ∂ ∂ α ∫ 0 1 x α − 1 ( 1 − x ) β − 1 d x = 1 B ( α , β ) ∂ B ( α , β ) ∂ α = ∂ ln ⁡ B ( α , β ) ∂ α = ∂ ln ⁡ Γ ( α ) ∂ α − ∂ ln ⁡ Γ ( α + β...

and f′(x) ≠ 0 for x ≠ α, x in a neighborhood U of α, α being a zero of multiplicity r, and if f ∈ Cr(U), then there exists a neighborhood of α such that...

order α {\displaystyle \alpha } , where 0 < α < ∞ {\displaystyle 0<\alpha <\infty } and α ≠ 1 {\displaystyle \alpha \neq 1} , is defined as H α ( X ) =...

function max ( x 1 , … , x n ) , {\displaystyle \max(x_{1},\ldots ,x_{n}),} meaning a parametric family of functions m α ( x 1 , … , x n ) {\displaystyle...

parameterization is f ( x ; α , β ) = x α − 1 e − β x β α Γ ( α )  for  x > 0 α , β > 0 , {\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}e^{-\beta...

defined as follows: α {\displaystyle \alpha } (or φ {\displaystyle \varphi } ) is the signed angle between the x axis and the N axis (x-convention – it could...

constants α such that a 0 e α x + a 1 α e α x + a 2 α 2 e α x + ⋯ + a n α n e α x = 0. {\displaystyle a_{0}e^{\alpha x}+a_{1}\alpha e^{\alpha x}+a_{2}\alpha...

x , y ) = x α y 1 − α {\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }} the marginal utility of x {\displaystyle x\,} is U 1 ( x , y ) = α (...

set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α : α ∈ A...

) x s + α − 1 ( 1 − x ) f + β − 1 / B ( α , β ) ∫ y = 0 1 ( ( s + f s ) y s + α − 1 ( 1 − y ) f + β − 1 / B ( α , β ) ) d y = x s + α − 1 ( 1 − x ) f...

function M α X + β ( t ) = e β t M X ( α t ) {\displaystyle M_{\alpha X+\beta }(t)=e^{\beta t}M_{X}(\alpha t)} M α X + β ( t ) = E [ e ( α X + β ) t ]...

as: P = ∑ | α | ≤ m a α ( x ) D α   , {\displaystyle P=\sum _{|\alpha |\leq m}a_{\alpha }(x)D^{\alpha }\ ,} where α = ( α 1 , α 2 , ⋯ , α n ) {\displaystyle...

read α e ( x ) = x {\displaystyle \alpha _{e}(x)=x} and α g ( α h ( x ) ) = ( α g ∘ α h ) ( x ) = α g h ( x ) {\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha...

⁡ [ X i ] = α i α 0 , {\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}},} Var ⁡ [ X i ] = α i ( α 0 − α i ) α 0 2 ( α 0 + 1...

α ( x ) {\displaystyle e^{i\alpha (x)}} and e i α ′ ( x ) {\displaystyle e^{i\alpha '(x)}} is yet another symmetry transformation e i [ α ( x ) + α ′...

compound is X- α {\displaystyle \alpha } -gal (Xαgal, X-alpha-gal), or 5-Bromo-4-chloro-3-indolyl-α-D-galactopyranoside, which is hydrolyzed by α-galactosidase...

( x ) = P r ( X > x ) {\displaystyle P(x)=\mathrm {Pr} (X>x)} , P ( x ) = Pr ( X > x ) = C ∫ x ∞ p ( X ) d X = α − 1 x min − α + 1 ∫ x ∞ X − α d X = (...

the pair of equations: d x d t = α x − β x y , d y d t = δ x y − γ y , {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy,\\{\frac {dy}{dt}}&=\delta...

Fock states, | α ⟩ = e − | α | 2 2 ∑ n = 0 ∞ α n n ! | n ⟩ = e − | α | 2 2 e α a ^ † e − α ∗ a ^ | 0 ⟩ = e α a ^ † − α ∗ a ^ | 0 ⟩ = D ( α ) | 0 ⟩   , {\displaystyle...

Namely, x 2 + b a x + c a = ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β . {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=(x-\alpha )(x-\beta )=x^{2}-(\alpha...

= α Δ u . {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \Delta u.} Define a new function v ( t , x ) = u ( t / α , x ) {\displaystyle v(t,x)=u(t/\alpha...

( x ) = α n 2 H e n ( x α ) = ( α 2 ) n 2 H n ( x 2 α ) = e − α D 2 2 ( x n ) . {\displaystyle {\mathit {He}}_{n}^{[\alpha ]}(x)=\alpha ^{\frac {n}{2}}{\mathit...

( x ) } = f ~ α ( n ) = ∫ 0 ∞ e − x x α   L n α ( x )   f ( x )   d x {\displaystyle L\{f(x)\}={\tilde {f}}_{\alpha }(n)=\int _{0}^{\infty }e^{-x}x^{\alpha...