Question Number 104608 by ~blr237~ last updated on 22/Jul/20

  D^(1/2) (y)=1     ⇒   y(x)=(√(x/π))

Answered by OlafThorendsen last updated on 22/Jul/20

n∈N, ν∈C, (x^ν )^((n))  = ((Γ(ν+1))/(Γ(ν+1−n)))x^(ν−n)   By generalizing :  α∈Q, ν∈C, (x^ν )^((α))  = ((Γ(ν+1))/(Γ(ν+1−α)))x^(ν−α)   And for α = ν = (1/2) :  (x^(1/2) )^((1/2))  = ((Γ((3/2)))/(Γ(1)))x^0  = ((Γ((3/2)))/(Γ(1)))  with Γ(1) = 1 and Γ((3/2)) = ((√π)/2)  Then ((√x))^((1/2))  = ((√π)/2)  (2(√(x/π)))^((1/2))  = 1  Sorry sir I find y = 2(√(x/π))

Commented by~blr237~ last updated on 23/Jul/20

yes it′s correct : error of typo  But you didn′t prove the proposition only its interplay   That generalisation is also correct ( it′s the Caputo derivation) and that coincides with the Riemann-Liouville derivation when α=(1/2)   D^α (f)(x)=(1/(Γ(α))) (d/dx)(∫_(0 ) ^x t^(α−1) f(x−t)dt)      0<α  D^(1/2) (y)=1⇒D^(1/2) (D^(1/2) (y))=D^(1/2) (1)     ⇒ y′(x)= (1/(Γ((1/2)))) (d/dx)( 2(√x) )  ⇒y(x)=2(√(x/π))