1.17

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
1.17.E1	${\displaystyle{\displaystyle\delta\left(x\right)=0}}$ \Diracdelta@{x} = 0	${\displaystyle{\displaystyle x\neq 0}}$	Dirac(x) = 0	DiracDelta[x] == 0	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 1]
1.17.E2	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}\delta\left(x-a\right)\phi(x)\mathrm{d}x=\phi(a)}}$ \int_{-\infty}^{\infty}\Diracdelta@{x-a}\phi(x)\diff{x} = \phi(a)		int(Dirac(x - a)*phi(x), x = - infinity..infinity) = phi(a)	Integrate[DiracDelta[x - a]*\[Phi][x], {x, - Infinity, Infinity}, GenerateConditions->None] == \[Phi][a]	Successful	Successful	-	Successful [Tested: 10]
1.17.E6	${\displaystyle{\displaystyle\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\mathrm{d}x=\phi(a)}}$ \lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\diff{x} = \phi(a)		limit(sqrt((n)/(Pi))*int(exp(- n*(x - a)^(2))*phi(x), x = - infinity..infinity), n = infinity) = phi(a)	Limit[Sqrt[Divide[n,Pi]]*Integrate[Exp[- n*(x - a)^(2)]*\[Phi][x], {x, - Infinity, Infinity}, GenerateConditions->None], n -> Infinity, GenerateConditions->None] == \[Phi][a]	Successful	Aborted	-	Successful [Tested: 60]
1.17.E7	${\displaystyle{\displaystyle\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\mathrm{d}x=\tfrac{1}{2}\phi(a-)+\tfrac{1}{2}\phi(a+)}}$ \lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\diff{x} = \tfrac{1}{2}\phi(a-)+\tfrac{1}{2}\phi(a+)		limit(sqrt((n)/(Pi))*int(exp(- n*(x - a)^(2))*phi(x), x = - infinity..infinity), n = infinity) = (1)/(2)*phi(a -)+(1)/(2)*phi(a +)	Limit[Sqrt[Divide[n,Pi]]*Integrate[Exp[- n*(x - a)^(2)]*\[Phi][x], {x, - Infinity, Infinity}, GenerateConditions->None], n -> Infinity, GenerateConditions->None] == Divide[1,2]*\[Phi][a -]+Divide[1,2]*\[Phi][a +]	Error	Failure	-	Error
1.17.E8	${\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iat}\left(\int_{-\infty}^{\infty}\phi(x)e^{itx}\mathrm{d}x\right)\mathrm{d}t=\phi(a)}}$ \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iat}\left(\int_{-\infty}^{\infty}\phi(x)e^{itx}\diff{x}\right)\diff{t} = \phi(a)		(1)/(2*Pi)*int(exp(- I*a*t)*(int(phi(x)* exp(I*t*x), x = - infinity..infinity)), t = - infinity..infinity) = phi(a)	Divide[1,2*Pi]*Integrate[Exp[- I*a*t]*(Integrate[\[Phi][x]* Exp[I*t*x], {x, - Infinity, Infinity}, GenerateConditions->None]), {t, - Infinity, Infinity}, GenerateConditions->None] == \[Phi][a]	Failure	Aborted	Failed [60 / 60] Result: Float(undefined)+.7500000000*I Test Values: {a = -1.5, phi = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+1.299038106*I Test Values: {a = -1.5, phi = -1/2+1/2*I*3^(1/2)} Result: Float(undefined)-1.299038106*I Test Values: {a = -1.5, phi = 1/2-1/2*I*3^(1/2)} Result: Float(undefined)-.7500000000*I Test Values: {a = -1.5, phi = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Skipped - Because timed out
1.17.E9	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}\mathrm{d}t\right)\phi(x)\mathrm{d}x=\phi(a)}}$ \int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}\diff{t}\right)\phi(x)\diff{x} = \phi(a)		int(((1)/(2*Pi)*int(exp(I*(x - a)*t), t = - infinity..infinity))*phi(x), x = - infinity..infinity) = phi(a)	Integrate[(Divide[1,2*Pi]*Integrate[Exp[I*(x - a)*t], {t, - Infinity, Infinity}, GenerateConditions->None])*\[Phi][x], {x, - Infinity, Infinity}, GenerateConditions->None] == \[Phi][a]	Successful	Aborted	-	Skipped - Because timed out
1.17.E10	${\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-t^{2}/(4n)}e^{i(x-a)t}\mathrm{d}t=\sqrt{\frac{n}{\pi}}e^{-n(x-a)^{2}}}}$ \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-t^{2}/(4n)}e^{i(x-a)t}\diff{t} = \sqrt{\frac{n}{\pi}}e^{-n(x-a)^{2}}		(1)/(2*Pi)*int(exp(- (t)^(2)/(4*n))*exp(I*(x - a)*t), t = - infinity..infinity) = sqrt((n)/(Pi))*exp(- n*(x - a)^(2))	Divide[1,2*Pi]*Integrate[Exp[- (t)^(2)/(4*n)]*Exp[I*(x - a)*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Sqrt[Divide[n,Pi]]*Exp[- n*(x - a)^(2)]	Failure	Successful	Successful [Tested: 54]	Successful [Tested: 54]
1.17.E12	${\displaystyle{\displaystyle\delta\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}\mathrm{d}t}}$ \Diracdelta@{x-a} = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}\diff{t}		Dirac(x - a) = (1)/(2*Pi)*int(exp(I*(x - a)*t), t = - infinity..infinity)	DiracDelta[x - a] == Divide[1,2*Pi]*Integrate[Exp[I*(x - a)*t], {t, - Infinity, Infinity}, GenerateConditions->None]	Successful	Failure	-	Skipped - Because timed out
1.17.E13	${\displaystyle{\displaystyle\delta\left(x-a\right)=x\int_{0}^{\infty}tJ_{\nu}\left(xt\right)J_{\nu}\left(at\right)\mathrm{d}t}}$ \Diracdelta@{x-a} = x\int_{0}^{\infty}t\BesselJ{\nu}@{xt}\BesselJ{\nu}@{at}\diff{t}	${\displaystyle{\displaystyle\Re\nu>-1,x>0,a>0}}$	Dirac(x - a) = x*int(t*BesselJ(nu, x*t)*BesselJ(nu, a*t), t = 0..infinity)	DiracDelta[x - a] == x*Integrate[t*BesselJ[\[Nu], x*t]*BesselJ[\[Nu], a*t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Error	Skipped - Because timed out
1.17.E14	${\displaystyle{\displaystyle\delta\left(x-a\right)=\frac{2xa}{\pi}\int_{0}^{\infty}t^{2}\mathsf{j}_{\ell}\left(xt\right)\mathsf{j}_{\ell}\left(at\right)\mathrm{d}t}}$ \Diracdelta@{x-a} = \frac{2xa}{\pi}\int_{0}^{\infty}t^{2}\sphBesselJ{\ell}@{xt}\sphBesselJ{\ell}@{at}\diff{t}	${\displaystyle{\displaystyle x>0,a>0}}$	Error	DiracDelta[x - a] == Divide[2*x*a,Pi]*Integrate[(t)^(2)* SphericalBesselJ[\[ScriptL], x*t]*SphericalBesselJ[\[ScriptL], a*t], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
1.17.E16	${\displaystyle{\displaystyle\delta\left(x-a\right)=\int_{-\infty}^{\infty}\mathrm{Ai}\left(t-x\right)\mathrm{Ai}\left(t-a\right)\mathrm{d}t}}$ \Diracdelta@{x-a} = \int_{-\infty}^{\infty}\AiryAi@{t-x}\AiryAi@{t-a}\diff{t}		Dirac(x - a) = int(AiryAi(t - x)*AiryAi(t - a), t = - infinity..infinity)	DiracDelta[x - a] == Integrate[AiryAi[t - x]*AiryAi[t - a], {t, - Infinity, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
1.17.E17	${\displaystyle{\displaystyle\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{-ika}\left(\int_{-\pi}^{\pi}\phi(x)e^{ikx}\mathrm{d}x\right)=\phi(a)}}$ \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{-ika}\left(\int_{-\pi}^{\pi}\phi(x)e^{ikx}\diff{x}\right) = \phi(a)		(1)/(2*Pi)*sum(exp(- I*k*a)*(int(phi(x)* exp(I*k*x), x = - Pi..Pi)), k = - infinity..infinity) = phi(a)	Divide[1,2*Pi]*Sum[Exp[- I*k*a]*(Integrate[\[Phi][x]* Exp[I*k*x], {x, - Pi, Pi}, GenerateConditions->None]), {k, - Infinity, Infinity}, GenerateConditions->None] == \[Phi][a]	Error	Failure	-	Successful [Tested: 60]
1.17.E18	${\displaystyle{\displaystyle\int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{ik(x-a)}\right)\mathrm{d}x=\phi(a)}}$ \int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{ik(x-a)}\right)\diff{x} = \phi(a)		int(phi(x)*((1)/(2*Pi)*sum(exp(I*k*(x - a)), k = - infinity..infinity)), x = - Pi..Pi) = phi(a)	Integrate[\[Phi][x]*(Divide[1,2*Pi]*Sum[Exp[I*k*(x - a)], {k, - Infinity, Infinity}, GenerateConditions->None]), {x, - Pi, Pi}, GenerateConditions->None] == \[Phi][a]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
1.17.E21	${\displaystyle{\displaystyle\delta\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{ik(x-a)}}}$ \Diracdelta@{x-a} = \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{ik(x-a)}		Dirac(x - a) = (1)/(2*Pi)*sum(exp(I*k*(x - a)), k = - infinity..infinity)	DiracDelta[x - a] == Divide[1,2*Pi]*Sum[Exp[I*k*(x - a)], {k, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [18 / 18] Result: Times[-0.15915494309189535, NSum[Power[E, Times[Complex[0.0, 3.0], k]] Test Values: {k, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 1.5]} Result: Times[-0.15915494309189535, NSum[Power[E, Times[Complex[0.0, 2.0], k]] Test Values: {k, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 0.5]} ... skip entries to safe data
1.17.E22	${\displaystyle{\displaystyle\delta\left(x-a\right)=\sum_{k=0}^{\infty}(k+\tfrac{1}{2})P_{k}\left(x\right)P_{k}\left(a\right)}}$ \Diracdelta@{x-a} = \sum_{k=0}^{\infty}(k+\tfrac{1}{2})\LegendrepolyP{k}@{x}\LegendrepolyP{k}@{a}		Dirac(x - a) = sum((k +(1)/(2))*LegendreP(k, x)*LegendreP(k, a), k = 0..infinity)	DiracDelta[x - a] == Sum[(k +Divide[1,2])*LegendreP[k, x]*LegendreP[k, a], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [18 / 18] Result: Times[-1.0, NSum[Times[Plus[Rational[1, 2], k], LegendreP[k, -1.5], LegendreP[k, 1.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 1.5]} Result: Times[-1.0, NSum[Times[Plus[Rational[1, 2], k], LegendreP[k, -1.5], LegendreP[k, 0.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 0.5]} ... skip entries to safe data
1.17.E23	${\displaystyle{\displaystyle\delta\left(x-a\right)=e^{-(x+a)/2}\sum_{k=0}^{\infty}L_{k}\left(x\right)L_{k}\left(a\right)}}$ \Diracdelta@{x-a} = e^{-(x+a)/2}\sum_{k=0}^{\infty}\LaguerrepolyL[]{k}@{x}\LaguerrepolyL[]{k}@{a}		Dirac(x - a) = exp(-(x + a)/2)*sum(LaguerreL(k, x)*LaguerreL(k, a), k = 0..infinity)	DiracDelta[x - a] == Exp[-(x + a)/2]*Sum[LaguerreL[k, x]*LaguerreL[k, a], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [18 / 18] Result: Times[-1.0, NSum[Times[LaguerreL[k, -1.5], LaguerreL[k, 1.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 1.5]} Result: Times[-1.6487212707001282, NSum[Times[LaguerreL[k, -1.5], LaguerreL[k, 0.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 0.5]} ... skip entries to safe data
1.17.E24	${\displaystyle{\displaystyle\delta\left(x-a\right)=\frac{e^{-(x^{2}+a^{2})/2}}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{H_{k}\left(x\right)H_{k}\left(a\right)}{2^{k}k!}}}$ \Diracdelta@{x-a} = \frac{e^{-(x^{2}+a^{2})/2}}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{\HermitepolyH{k}@{x}\HermitepolyH{k}@{a}}{2^{k}k!}		Dirac(x - a) = (exp(-((x)^(2)+ (a)^(2))/2))/(sqrt(Pi))*sum((HermiteH(k, x)*HermiteH(k, a))/((2)^(k)* factorial(k)), k = 0..infinity)	DiracDelta[x - a] == Divide[Exp[-((x)^(2)+ (a)^(2))/2],Sqrt[Pi]]*Sum[Divide[HermiteH[k, x]*HermiteH[k, a],(2)^(k)* (k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [18 / 18] Result: Times[-0.05946514461181468, NSum[Times[Power[2, Times[-1, k]], Power[Factorial[k], -1], HermiteH[k, -1.5], HermiteH[k, 1.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 1.5]} Result: Times[-0.16164302202498515, NSum[Times[Power[2, Times[-1, k]], Power[Factorial[k], -1], HermiteH[k, -1.5], HermiteH[k, 0.5]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[a, -1.5], Rule[x, 0.5]} ... skip entries to safe data
1.17.E25	${\displaystyle{\displaystyle\delta\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta\left(\phi_{1}-\phi_{2}\right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{{\ell},{m}}\left(\theta_{1},\phi_{1}\right)\overline{Y_{{\ell},{m}}\left(\theta_{2},\phi_{2}\right)}}}$ \Diracdelta@{\cos@@{\theta_{1}}-\cos@@{\theta_{2}}}\Diracdelta@{\phi_{1}-\phi_{2}} = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\sphharmonicY{\ell}{m}@{\theta_{1}}{\phi_{1}}\conj{\sphharmonicY{\ell}{m}@{\theta_{2}}{\phi_{2}}}		Dirac(cos(theta[1])- cos(theta[2]))*Dirac(phi[1]- phi[2]) = sum(sum(SphericalY(ell, m, theta[1], phi[1])*conjugate(SphericalY(ell, m, theta[2], phi[2])), m = - ell..ell), ell = 0..infinity)	DiracDelta[Cos[Subscript[\[Theta], 1]]- Cos[Subscript[\[Theta], 2]]]*DiracDelta[Subscript[\[Phi], 1]- Subscript[\[Phi], 2]] == Sum[Sum[SphericalHarmonicY[\[ScriptL], m, Subscript[\[Theta], 1], Subscript[\[Phi], 1]]*Conjugate[SphericalHarmonicY[\[ScriptL], m, Subscript[\[Theta], 2], Subscript[\[Phi], 2]]], {m, - \[ScriptL], \[ScriptL]}, GenerateConditions->None], {\[ScriptL], 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Skipped - Because timed out	Skipped - Because timed out