1.16

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
1.16.E1	${\displaystyle{\displaystyle\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2})=\alpha_{1}\Lambda(\phi_{1})+\alpha_{2}\Lambda(\phi_{2})}}$ \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	${\displaystyle{\displaystyle\lim_{n\to\infty}\Lambda(\phi_{n})=\Lambda(\phi)}}$ \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	${\displaystyle{\displaystyle\sigma_{n}=f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-)}}$ \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	${\displaystyle{\displaystyle x^{\alpha}_{+}=x^{\alpha}H\left(x\right)}}$ 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	${\displaystyle{\displaystyle Dx^{\alpha}_{+}=\alpha x_{+}^{\alpha-1}}}$ 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	${\displaystyle{\displaystyle x^{\alpha}_{+}=\frac{1}{(\alpha+1)_{n}}D^{n}x_{+}^{\alpha+n}}}$ 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	${\displaystyle{\displaystyle\ln_{+}x=H\left(x\right)\ln x}}$ \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	${\displaystyle{\displaystyle(-1)^{n}n!x_{+}^{-1-n}=D^{(n+1)}\ln_{+}x}}$ (-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	${\displaystyle{\displaystyle|x^{N}\phi_{n}^{(k)}|\leq c_{k,N}}}$ |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	${\displaystyle{\displaystyle|x^{m}\phi^{(k)}(x)|\leq c_{m,k}}}$ |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	${\displaystyle{\displaystyle D_{\boldsymbol{{\alpha}}}={\mathrm{i}^{-|\boldsymbol{{\alpha}}|}}D^{\boldsymbol{{\alpha}}}}}$ 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	${\displaystyle{\displaystyle P(\mathbf{x})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{x}^{\boldsymbol{{\alpha}}}}}$ 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	${\displaystyle{\displaystyle P(D)=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}D_{\boldsymbol{{\alpha}}}}}$ 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	${\displaystyle{\displaystyle\mathcal{F}(P(D)u)=P(-\mathbf{x})\mathcal{F}(u)}}$ \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	${\displaystyle{\displaystyle\mathcal{F}(Pu)=P(D)\mathcal{F}(u)}}$ \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	${\displaystyle{\displaystyle\int^{\infty}_{-\infty}\delta\left(t\right){\mathrm{e}^{\mathrm{i}xt}}\mathrm{d}t=1}}$ \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	${\displaystyle{\displaystyle\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}^{\mathrm{i}xt}}\mathrm{d}t=\delta\left(x\right)}}$ \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	${\displaystyle{\displaystyle\operatorname{sign}\left(x\right)=2H\left(x\right)-1}}$ \sign@{x} = 2\HeavisideH@{x}-1	${\displaystyle{\displaystyle x\neq 0}}$	signum(x) = 2*Heaviside(x)- 1	Sign[x] == 2*HeavisideTheta[x]- 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]