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 ) = ( 2 + α − 1 − x n ) L n − 1 ( α ) ( x ) − ( 1 + α − 1 n ) L n − 2 ( α ) ( x ) = α + 1 − x n L n − 1 ( α + 1 ) ( x ) − x n L n − 2 ( α +...

= 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...

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...

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 ≥...

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...

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...

) 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...

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...

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...

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 ]...

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

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 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...

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...

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 ′...

= 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 ) = 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 = (...

( 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...

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...

reconciliations can be useful: ES α ⁡ ( X ) = − 1 α E ⁡ [ X ] + 1 − α α ES α ⁡ ( L )  and  ES α ⁡ ( L ) = 1 1 − α E ⁡ [ L ] + α 1 − α ES α ⁡ ( X ) . {\displaystyle \operatorname...

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...

T = x sinh ⁡ ( α τ ) , X = x cosh ⁡ ( α τ ) {\displaystyle T=x\sinh(\alpha \tau ),\quad X=x\cosh(\alpha \tau )} where x = 1 / α {\displaystyle x=1/\alpha...

Then Wα(x) is Hölder continuous of exponent α, which is to say that there is a constant C such that | W α ( x ) − W α ( y ) | ≤ C | x − y | α {\displaystyle...

dynamical variables are a collection of functions {φα}α over a d-dimensional space/spacetime manifold M where α is the "flavor" index. This involves functionals...