2

2+2

x+x

2 2

656-360x+(((2x-3))^2)+16(x^3)/.(((x_))^2)->(((x-5))^3)

Plot[656-360 x+16 (x^3)+(((-8+2 x))^3),{x,-2,5}]

a(b+c)
a[b+c]

p[x_]:=656-360 x+16 (x^3)+(((-8+2 x))^3)

Plot[p[x],{x,-2,5}]

β[x_,m_,n_]:=Module[{ϛ,ξ,α,k},ϛ[ξ_,α_]:=(1/(2α!))Sign[ξ](ξ^α);(∑_(k=0)^(m+1))(((-1))^k)Binomial[m+1,k]ϛ[x+((m+1)/2)-k,m-n]]

TableForm[Table[Plot[β[x,n,m],{x,-3,3}],{n,0,4},{m,0,n}]]

?`p

p[x_]=.

?`*
?`p

Clear[p]

?`*
?`p

Remove[p]

?`*
?`p

p[x]=656-360 x+16 (x^3)+(((-8+2 x))^3)

p[x]

Plot[p[x],{x,-2,5}]

p[y]

p[x_]=656-360 x+16 (x^3)+(((-8+2 x))^3)

p[y+z]

Plot[p[ξ],{ξ,-2,5}]

f[0]=1
f[n_]=n f[n-1]

f[5]

f[0]=1
f[n_]:=n f[n-1]

f[5]

g=(1/(σ(Sqrt[2π])))(E^(-(((#)^2)/(2(σ^2)))))&

g[0]
g[ σ(Sqrt[Log[4]])]
g[x-y]

Plot[g[x]/.σ->1,{x,-5,5}]

Derivative[4][g]

(x+1)(x-1)

Simplify[(x+1)(x-1)]

Simplify[Cos[θ]+I Sin[θ]]

FullSimplify[Cos[θ]+I Sin[θ]]

Simplify[Derivative[4][g]]

Simplify[Derivative[4][g][x]]

uniform=Table[RandomReal[],{3}]

uniform[[0]]

uniform[[1]]
uniform[[2]]
uniform[[3]]

gaussian=RandomVariate[NormalDistribution[0,1],3]

m=Table[{i,j},{i,1,5},{j,1,3}]

TableForm[m]

MatrixForm[m]

m[[2,3]]

m=Table[(Sqrt[(2/(#))])Sin[((2i j π)/(#))+(π/4)],{i,0,#-1},{j,0,#-1}]&[3]

gaussian
m.gaussian
m.m.gaussian

Polynomial B-Splines

derivative[f_,x_,n_]:=Module[{y},D[f[y],{y,n}]/.y->x]

continuousImpulseResponse[x_,n_]:=derivative[DiracDelta,x,n]

(OverHat[h])[ω_,n_]:=Module[{x,Ω},FourierTransform[continuousImpulseResponse[x,n],x,Ω,FourierParameters->{1, -1}]/.Ω->ω]

finiteDifference[f_,x_,0]:=f[x]
finiteDifference[f_,x_,n_]:=finiteDifference[f[#+(1/2)]-f[#-(1/2)]&,x,n-1]

discreteImpulseResponse[x_,n_]:=finiteDifference[DiracDelta,x,n]

(OverHat[H])[ω_,n_]:=Module[{x,Ω},FourierTransform[discreteImpulseResponse[x,n],x,Ω,FourierParameters->{1, -1}]/.Ω->ω]

β[x_,n_]:=Module[{y,ω},InverseFourierTransform[(((OverHat[H])[ω,n+1])/((OverHat[h])[ω,n+1])),ω,y,FourierParameters->{1,-1}]/.y->x]

β[x,5]

Plot[(1/240) (-15 (((-1+x))^5) Sign[1-x]+6 (((-2+x))^5) Sign[2-x]-(((-3+x))^5) Sign[3-x]-20 (x^5) Sign[x]+15 (((1+x))^5) Sign[1+x]-6 (((2+x))^5) Sign[2+x]+(((3+x))^5) Sign[3+x]),{x,-4,4},PlotRange->All]