3.11

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
3.11.E5	${\displaystyle{\displaystyle\sum_{k=0}^{n}x_{j}^{k}\delta a_{k}=(-1)^{j}(m_{j}-m)}}$ \sum_{k=0}^{n}x_{j}^{k}\delta a_{k} = (-1)^{j}(m_{j}-m)	${\displaystyle{\displaystyle j=0}}$	sum((x[j])^(k)*delta*a[k], k = 0..n) = (- 1)^(j)*(((- 1)^(j)* epsilon[n](x[j]))- m)	Sum[(Subscript[x, j])^(k)*\[Delta]*Subscript[a, k], {k, 0, n}, GenerateConditions->None] == (- 1)^(j)*(((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]])- m)	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E6	${\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\operatorname{arccos}x\right)}}$ \ChebyshevpolyT{n}@{x} = \cos@{n\acos@@{x}}	${\displaystyle{\displaystyle-1\leq x,x\leq 1}}$	ChebyshevT(n, x) = cos(n*arccos(x))	ChebyshevT[n, x] == Cos[n*ArcCos[x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
3.11.E7	${\displaystyle{\displaystyle T_{n+1}\left(x\right)-2xT_{n}\left(x\right)+T_{n-1}\left(x\right)=0}}$ \ChebyshevpolyT{n+1}@{x}-2x\ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n-1}@{x} = 0		ChebyshevT(n + 1, x)- 2*x*ChebyshevT(n, x)+ ChebyshevT(n - 1, x) = 0	ChebyshevT[n + 1, x]- 2*x*ChebyshevT[n, x]+ ChebyshevT[n - 1, x] == 0	Successful	Successful	-	Successful [Tested: 3]
3.11.E13	${\displaystyle{\displaystyle\epsilon_{n}(x)=d_{n+1}T_{n+1}\left(\frac{2x-a-b}{b-a}\right)}}$ \epsilon_{n}(x) = d_{n+1}\ChebyshevpolyT{n+1}@{\frac{2x-a-b}{b-a}}		epsilon[n](x) = d[n + 1]*ChebyshevT(n + 1, (2*x - a - b)/(b - a))	Subscript[\[Epsilon], n][x] == Subscript[d, n + 1]*ChebyshevT[n + 1, Divide[2*x - a - b,b - a]]	Failure	Failure	Failed [300 / 300] Result: Float(-infinity)-Float(infinity)*I Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 1} Result: Float(-infinity)-Float(infinity)*I Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 2} Result: Float(-infinity)-Float(infinity)*I Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 3} Result: Float(-infinity)-Float(infinity)*I Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 2, n = 1} ... skip entries to safe data	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 1], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 2], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
3.11.E15	${\displaystyle{\displaystyle u_{k}=2xu_{k+1}-u_{k+2}+c_{k}}}$ u_{k} = 2xu_{k+1}-u_{k+2}+c_{k}	${\displaystyle{\displaystyle k=n-1}}$	u[k] = 2*x*u[k + 1]- u[k + 2]+ c[k]	Subscript[u, k] == 2*x*Subscript[u, k + 1]- Subscript[u, k + 2]+ Subscript[c, k]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E18	${\displaystyle{\displaystyle m_{j}=(-1)^{j}\epsilon_{k,\ell}(x_{j})}}$ m_{j} = (-1)^{j}\epsilon_{k,\ell}(x_{j})	${\displaystyle{\displaystyle j=0}}$	((- 1)^(j)* epsilon[n](x[j])) = (- 1)^(j)* epsilon[k , ell](x[j])	((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]]) == (- 1)^(j)* Subscript[\[Epsilon], k , \[ScriptL]][Subscript[x, j]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E19	${\displaystyle{\displaystyle R_{3,3}(x)=\frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}}}$ R_{3,3}(x) = \frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}		R[3 , 3](x) = (p[0]+ p[1]*x + p[2]*(x)^(2)+ p[3]*(x)^(3))/(1 + q[1]*x + q[2]*(x)^(2)+ q[3]*(x)^(3))	Subscript[R, 3 , 3][x] == Divide[Subscript[p, 0]+ Subscript[p, 1]*x + Subscript[p, 2]*(x)^(2)+ Subscript[p, 3]*(x)^(3),1 + Subscript[q, 1]*x + Subscript[q, 2]*(x)^(2)+ Subscript[q, 3]*(x)^(3)]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11#E23X	${\displaystyle{\displaystyle\displaystyle a_{0}=c_{0}b_{0}}}$ \displaystyle a_{0} = c_{0}b_{0}		a[0] = c[0]*b[0]	Subscript[a, 0] == Subscript[c, 0]*Subscript[b, 0]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11#E23Xa	${\displaystyle{\displaystyle\displaystyle a_{1}=c_{1}b_{0}+c_{0}b_{1}}}$ \displaystyle a_{1} = c_{1}b_{0}+c_{0}b_{1}		a[1] = c[1]*b[0]+ c[0]*b[1]	Subscript[a, 1] == Subscript[c, 1]*Subscript[b, 0]+ Subscript[c, 0]*Subscript[b, 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E25	${\displaystyle{\displaystyle(N-C)^{-1}+(S-C)^{-1}=(W-C)^{-1}+(E-C)^{-1}}}$ (N-C)^{-1}+(S-C)^{-1} = (W-C)^{-1}+(E-C)^{-1}		(N - C)^(- 1)+(S - C)^(- 1) = (W - C)^(- 1)+(E - C)^(- 1)	(N - C)^(- 1)+(S - C)^(- 1) == (W - C)^(- 1)+(E - C)^(- 1)	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E34	${\displaystyle{\displaystyle X_{k\ell}=\sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})\phi_{\ell}(x_{j})}}$ X_{k\ell} = \sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})\phi_{\ell}(x_{j})		X[k*ell] = sum(w(x[j])* phi[k](x[j])* phi[ell](x[j]), j = 1..J)	Subscript[X, k*\[ScriptL]] == Sum[w[Subscript[x, j]]* Subscript[\[Phi], k][Subscript[x, j]]* Subscript[\[Phi], \[ScriptL]][Subscript[x, j]], {j, 1, J}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E37	${\displaystyle{\displaystyle\sum_{j=0}^{n-1}\phi_{k}(x_{j})\overline{\phi_{\ell}(x_{j})}=n\delta_{k,\ell}}}$ \sum_{j=0}^{n-1}\phi_{k}(x_{j})\conj{\phi_{\ell}(x_{j})} = n\Kroneckerdelta{k}{\ell}	${\displaystyle{\displaystyle k=0,\ell=0}}$	sum(phi[k](x[j])* conjugate(phi[ell](x[j])), j = 0..n - 1) = n*KroneckerDelta[k, ell]	Sum[Subscript[\[Phi], k][Subscript[x, j]]* Conjugate[Subscript[\[Phi], \[ScriptL]][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None] == n*KroneckerDelta[k, \[ScriptL]]	Error	Failure	-	Failed [300 / 300] Result: Plus[1.0, Times[-1.0, KroneckerDelta[1.0, ℓ]]] Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[2.0, Times[-2.0, KroneckerDelta[1.0, ℓ]]] Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
3.11.E38	${\displaystyle{\displaystyle f_{j}=\sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})}}$ f_{j} = \sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})	${\displaystyle{\displaystyle j=0}}$	f[j] = sum(a[k]*phi[k](x[j]), k = 0..n - 1)	Subscript[f, j] == Sum[Subscript[a, k]*Subscript[\[Phi], k][Subscript[x, j]], {k, 0, n - 1}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11.E39	${\displaystyle{\displaystyle a_{k}=\frac{1}{n}\sum_{j=0}^{n-1}f_{j}\overline{\phi_{k}(x_{j})}}}$ a_{k} = \frac{1}{n}\sum_{j=0}^{n-1}f_{j}\conj{\phi_{k}(x_{j})}	${\displaystyle{\displaystyle k=0}}$	a[k] = (1)/(n)*sum(f[j]*conjugate(phi[k](x[j])), j = 0..n - 1)	Subscript[a, k] == Divide[1,n]*Sum[Subscript[f, j]*Conjugate[Subscript[\[Phi], k][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None]	Error	Failure	-	Failed [300 / 300] Result: Complex[0.0, 0.9999999999999999] Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, 0.9999999999999999] Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
3.11#Ex5	${\displaystyle{\displaystyle f_{j}=\sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}}}$ f_{j} = \sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}		f[j] = sum(a[k]*(omega[n])^(j*k), k = 0..n - 1)	Subscript[f, j] == Sum[Subscript[a, k]*(Subscript[\[Omega], n])^(j*k), {k, 0, n - 1}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.11#Ex6	${\displaystyle{\displaystyle\omega_{n}=e^{2\pi i/n}}}$ \omega_{n} = e^{2\cpi i/n}	${\displaystyle{\displaystyle j=0}}$	omega[n] = exp(2*Pi*I/n)	Subscript[\[Omega], n] == Exp[2*Pi*I/n]	Failure	Failure	Failed [290 / 300] Result: -.1339745960+.4999999992*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 1.866025404+.5000000004*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 2} Result: 1.366025404-.3660254040*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 3} Result: -1.500000000+.8660254032*I Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = -1/2+1/2*I*3^(1/2), n = 1} ... skip entries to safe data	Failed [290 / 300] Result: Complex[-0.1339745962155613, 0.49999999999999994] Test Values: {Rule[n, 1], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.8660254037844388, 0.49999999999999994] Test Values: {Rule[n, 2], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
3.11.E42	${\displaystyle{\displaystyle\omega_{n}^{2(k-(n/2))}=\omega_{n/2}^{k}}}$ \omega_{n}^{2(k-(n/2))} = \omega_{n/2}^{k}		(omega[n])^(2*(k -(n/2))) = (omega[n/2])^(k)	(Subscript[\[Omega], n])^(2*(k -(n/2))) == (Subscript[\[Omega], n/2])^(k)	Skipped - no semantic math	Skipped - no semantic math	-	-