Algebraic and Analytic Methods - 1.16 Distributions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.16.E1 Λ ⁢ ( α 1 ⁢ ϕ 1 + α 2 ⁢ ϕ 2 ) = α 1 ⁢ Λ ⁢ ( ϕ 1 ) + α 2 ⁢ Λ ⁢ ( ϕ 2 ) Λ subscript 𝛼 1 subscript italic-ϕ 1 subscript 𝛼 2 subscript italic-ϕ 2 subscript 𝛼 1 Λ subscript italic-ϕ 1 subscript 𝛼 2 Λ subscript italic-ϕ 2 {\displaystyle{\displaystyle\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2})=% \alpha_{1}\Lambda(\phi_{1})+\alpha_{2}\Lambda(\phi_{2})}}
\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2}) = \alpha_{1}\Lambda(\phi_{1})+\alpha_{2}\Lambda(\phi_{2})		 

Lambda(alpha[1]*phi[1]+ alpha[2]*phi[2]) = alpha[1]*Lambda(phi[1])+ alpha[2]*Lambda(phi[2])
 
\[CapitalLambda][Subscript[\[Alpha], 1]*Subscript[\[Phi], 1]+ Subscript[\[Alpha], 2]*Subscript[\[Phi], 2]] == Subscript[\[Alpha], 1]*\[CapitalLambda][Subscript[\[Phi], 1]]+ Subscript[\[Alpha], 2]*\[CapitalLambda][Subscript[\[Phi], 2]]
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E2 lim n → ∞ ⁡ Λ ⁢ ( ϕ n ) = Λ ⁢ ( ϕ ) subscript → 𝑛 Λ subscript italic-ϕ 𝑛 Λ italic-ϕ {\displaystyle{\displaystyle\lim_{n\to\infty}\Lambda(\phi_{n})=\Lambda(\phi)}}
\lim_{n\to\infty}\Lambda(\phi_{n}) = \Lambda(\phi)		 

limit(Lambda(phi[n]), n = infinity) = Lambda(phi)
 
Limit[\[CapitalLambda][Subscript[\[Phi], n]], n -> Infinity, GenerateConditions->None] == \[CapitalLambda][\[Phi]]
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E17 σ n = f ( n ) ⁢ ( x 0 + ) - f ( n ) ⁢ ( x 0 - ) subscript 𝜎 𝑛 superscript 𝑓 𝑛 limit-from subscript 𝑥 0 superscript 𝑓 𝑛 limit-from subscript 𝑥 0 {\displaystyle{\displaystyle\sigma_{n}=f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-)}}
\sigma_{n} = f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-)		 

sigma[n] = (f)^(n)*(x[0]+)- (f)^(n)*(x[0]-)
 
Subscript[\[Sigma], n] == (f)^(n)*(Subscript[x, 0]+)- (f)^(n)*(Subscript[x, 0]-)
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E19 x + α = x α ⁢ H ⁡ ( x ) subscript superscript 𝑥 𝛼 superscript 𝑥 𝛼 Heaviside-H 𝑥 {\displaystyle{\displaystyle x^{\alpha}_{+}=x^{\alpha}H\left(x\right)}}
x^{\alpha}_{+} = x^{\alpha}\HeavisideH@{x}		 

(x[+])^(alpha) = (x)^(alpha)* Heaviside(x)

(Subscript[x, +])^\[Alpha] == (x)^\[Alpha]* HeavisideTheta[x]
Error Failure - Error
1.16.E20 D ⁢ x + α = α ⁢ x + α - 1 𝐷 subscript superscript 𝑥 𝛼 𝛼 superscript subscript 𝑥 𝛼 1 {\displaystyle{\displaystyle Dx^{\alpha}_{+}=\alpha x_{+}^{\alpha-1}}}
Dx^{\alpha}_{+} = \alpha x_{+}^{\alpha-1}		 

D*(x[+])^(alpha) = alpha*(x[+])^(alpha - 1)
 
D*(Subscript[x, +])^\[Alpha] == \[Alpha]*(Subscript[x, +])^(\[Alpha]- 1)
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E21 x + α = 1 ( α + 1 ) n ⁢ D n ⁢ x + α + n subscript superscript 𝑥 𝛼 1 subscript 𝛼 1 𝑛 superscript 𝐷 𝑛 superscript subscript 𝑥 𝛼 𝑛 {\displaystyle{\displaystyle x^{\alpha}_{+}=\frac{1}{(\alpha+1)_{n}}D^{n}x_{+}% ^{\alpha+n}}}
x^{\alpha}_{+} = \frac{1}{(\alpha+1)_{n}}D^{n}x_{+}^{\alpha+n}		 

(x[+])^(alpha) = (1)/(alpha + 1[n])*(D)^(n)* (x[+])^(alpha + n)
 
(Subscript[x, +])^\[Alpha] == Divide[1,Subscript[\[Alpha]+ 1, n]]*(D)^(n)* (Subscript[x, +])^(\[Alpha]+ n)
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E22 ln + ⁡ x = H ⁡ ( x ) ⁢ ln ⁡ x subscript 𝑥 Heaviside-H 𝑥 𝑥 {\displaystyle{\displaystyle\ln_{+}x=H\left(x\right)\ln x}}
\ln_{+}x = \HeavisideH@{x}\ln@@{x}		 

$0[+]ln()*x = Heaviside(x)*ln(x)

Subscript[$0, +]Log[]*x == HeavisideTheta[x]*Log[x]
Translation Error Translation Error - -
1.16.E23 ( - 1 ) n ⁢ n ! ⁢ x + - 1 - n = D ( n + 1 ) ⁢ ln + ⁡ x superscript 1 𝑛 𝑛 superscript subscript 𝑥 1 𝑛 superscript 𝐷 𝑛 1 subscript 𝑥 {\displaystyle{\displaystyle(-1)^{n}n!x_{+}^{-1-n}=D^{(n+1)}\ln_{+}x}}
(-1)^{n}n!x_{+}^{-1-n} = D^{(n+1)}\ln_{+}x		 

(- 1)^(n)* factorial(n)*(x[+])^(- 1 - n) = (D)^(n + 1)* [+]ln()*x
 
(- 1)^(n)* (n)!*(Subscript[x, +])^(- 1 - n) == Subscript[(D)^(n + 1)* , +]Log[]*x
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E24 | x N ⁢ ϕ n ( k ) | ≤ c k , N superscript 𝑥 𝑁 superscript subscript italic-ϕ 𝑛 𝑘 subscript 𝑐 𝑘 𝑁 {\displaystyle{\displaystyle|x^{N}\phi_{n}^{(k)}|\leq c_{k,N}}}
|x^{N}\phi_{n}^{(k)}| \leq c_{k,N}		 

abs((x)^(N)* (phi[n])^(k)) <= c[k , N]
 
Abs[(x)^(N)* (Subscript[\[Phi], n])^(k)] <= Subscript[c, k , N]
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E28 | x m ⁢ ϕ ( k ) ⁢ ( x ) | ≤ c m , k superscript 𝑥 𝑚 superscript italic-ϕ 𝑘 𝑥 subscript 𝑐 𝑚 𝑘 {\displaystyle{\displaystyle|x^{m}\phi^{(k)}(x)|\leq c_{m,k}}}
|x^{m}\phi^{(k)}(x)| \leq c_{m,k}		 

abs((x)^(m)* (phi(x)*)^(k)) <= c[m , k]
 
Abs[(x)^(m)* (\[Phi][x]*)^(k)] <= Subscript[c, m , k]
 Skipped - no semantic math Skipped - no semantic math - -
1.16#Ex2 D 𝜶 = i - | 𝜶 | ⁢ D 𝜶 subscript 𝐷 𝜶 imaginary-unit 𝜶 superscript 𝐷 𝜶 {\displaystyle{\displaystyle D_{\boldsymbol{{\alpha}}}={\mathrm{i}^{-|% \boldsymbol{{\alpha}}|}}D^{\boldsymbol{{\alpha}}}}}
D_{\boldsymbol{{\alpha}}} = \iunit^{-|\boldsymbol{{\alpha}}|}D^{\boldsymbol{{\alpha}}}		 

D[alpha] = (I)^(-abs(alpha))* (D)^(alpha)

Subscript[D, \[Alpha]] == (I)^(-Abs[\[Alpha]])* (D)^\[Alpha]
Failure Failure
Failed [300 / 300]
Result: .8660254041+1.500000000*I
Test Values: {D = 1/2*3^(1/2)+1/2*I, alpha = 1.5, D[alpha] = 1/2*3^(1/2)+1/2*I}


Result: -.4999999999+1.866025404*I
Test Values: {D = 1/2*3^(1/2)+1/2*I, alpha = 1.5, D[alpha] = -1/2+1/2*I*3^(1/2)}


Result: .5000000001+.1339745960*I
Test Values: {D = 1/2*3^(1/2)+1/2*I, alpha = 1.5, D[alpha] = 1/2-1/2*I*3^(1/2)}


Result: -.8660254039+.5000000000*I
Test Values: {D = 1/2*3^(1/2)+1/2*I, alpha = 1.5, D[alpha] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.8660254037844387, 1.5]
Test Values: {Rule[D, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[D, α], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4999999999999998, 1.8660254037844388]
Test Values: {Rule[D, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[Subscript[D, α], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.16.E31 P ⁢ ( 𝐱 ) = ∑ 𝜶 c 𝜶 ⁢ 𝐱 𝜶 𝑃 𝐱 subscript 𝜶 subscript 𝑐 𝜶 superscript 𝐱 𝜶 {\displaystyle{\displaystyle P(\mathbf{x})=\sum_{\boldsymbol{{\alpha}}}c_{% \boldsymbol{{\alpha}}}\mathbf{x}^{\boldsymbol{{\alpha}}}}}
P(\mathbf{x}) = \sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{x}^{\boldsymbol{{\alpha}}}		 

P(x) = sum(c[alpha]*(x)^(alpha), $0[alpha]*c[alpha]*(x)^(alpha) = - infinity..infinity)
 
P[x] == Sum[Subscript[c, \[Alpha]]*(x)^\[Alpha], {Subscript[$0, \[Alpha]]*Subscript[c, \[Alpha]]*(x)^\[Alpha], - Infinity, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.16#Ex3 P ⁢ ( D ) = ∑ 𝜶 c 𝜶 ⁢ D 𝜶 𝑃 𝐷 subscript 𝜶 subscript 𝑐 𝜶 subscript 𝐷 𝜶 {\displaystyle{\displaystyle P(D)=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{% \alpha}}}D_{\boldsymbol{{\alpha}}}}}
P(D) = \sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}D_{\boldsymbol{{\alpha}}}		 

P(D) = sum(c[alpha]*D[alpha], $0[alpha]*c[alpha]*D[alpha] = - infinity..infinity)
 
P[D] == Sum[Subscript[c, \[Alpha]]*Subscript[D, \[Alpha]], {Subscript[$0, \[Alpha]]*Subscript[c, \[Alpha]]*Subscript[D, \[Alpha]], - Infinity, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.16#Ex7 ℱ ⁢ ( P ⁢ ( D ) ⁢ u ) = P ⁢ ( - 𝐱 ) ⁢ ℱ ⁢ ( u ) ℱ 𝑃 𝐷 𝑢 𝑃 𝐱 ℱ 𝑢 {\displaystyle{\displaystyle\mathcal{F}(P(D)u)=P(-\mathbf{x})\mathcal{F}(u)}}
\mathcal{F}(P(D)u) = P(-\mathbf{x})\mathcal{F}(u)		 

F(P(D)* u) = P(- x)* F(u)
 
F[P[D]* u] == P[- x]* F[u]
 Skipped - no semantic math Skipped - no semantic math - -
1.16#Ex8 ℱ ⁢ ( P ⁢ u ) = P ⁢ ( D ) ⁢ ℱ ⁢ ( u ) ℱ 𝑃 𝑢 𝑃 𝐷 ℱ 𝑢 {\displaystyle{\displaystyle\mathcal{F}(Pu)=P(D)\mathcal{F}(u)}}
\mathcal{F}(Pu) = P(D)\mathcal{F}(u)		 

F(P*u) = P(D)* F(u)
 
F[P*u] == P[D]* F[u]
 Skipped - no semantic math Skipped - no semantic math - -
1.16.E40 ∫ - ∞ ∞ δ ⁡ ( t ) ⁢ e i ⁢ x ⁢ t ⁢ d t = 1 subscript superscript Dirac-delta 𝑡 imaginary-unit 𝑥 𝑡 𝑡 1 {\displaystyle{\displaystyle\int^{\infty}_{-\infty}\delta\left(t\right){% \mathrm{e}^{\mathrm{i}xt}}\mathrm{d}t=1}}
\int^{\infty}_{-\infty}\Diracdelta@{t}\expe^{\iunit xt}\diff{t} = 1		 

int(Dirac(t)*exp(I*x*t), t = - infinity..infinity) = 1

Integrate[DiracDelta[t]*Exp[I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 3]
1.16.E43 1 2 ⁢ π ⁢ ∫ - ∞ ∞ e i ⁢ x ⁢ t ⁢ d t = δ ⁡ ( x ) 1 2 subscript superscript imaginary-unit 𝑥 𝑡 𝑡 Dirac-delta 𝑥 {\displaystyle{\displaystyle\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}^{% \mathrm{i}xt}}\mathrm{d}t=\delta\left(x\right)}}
\frac{1}{2\cpi}\int^{\infty}_{-\infty}\expe^{\iunit xt}\diff{t} = \Diracdelta@{x}		 

(1)/(2*Pi)*int(exp(I*x*t), t = - infinity..infinity) = Dirac(x)

Divide[1,2*Pi]*Integrate[Exp[I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None] == DiracDelta[x]
Successful Failure -
Failed [3 / 3]
Result: Complex[-0.07099199156997928, 3.003857199159988*^-16]
Test Values: {Rule[x, 1.5]}


Result: Complex[0.07742603591272186, 0.30312240144001046]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
1.16.E44 sign ⁡ ( x ) = 2 ⁢ H ⁡ ( x ) - 1 sign 𝑥 2 Heaviside-H 𝑥 1 {\displaystyle{\displaystyle\operatorname{sign}\left(x\right)=2H\left(x\right)% -1}}
\sign@{x} = 2\HeavisideH@{x}-1		 x ≠ 0 𝑥 0 {\displaystyle{\displaystyle x\neq 0}} 
signum(x) = 2*Heaviside(x)- 1

Sign[x] == 2*HeavisideTheta[x]- 1
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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