Algebraic and Analytic Methods - 1.17 Integral and Series Representations of the Dirac Delta

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.17.E1 Ξ΄ ⁑ ( x ) = 0 Dirac-delta π‘₯ 0 {\displaystyle{\displaystyle\delta\left(x\right)=0}}
\Diracdelta@{x} = 0		 x β‰  0 π‘₯ 0 {\displaystyle{\displaystyle x\neq 0}} 
Dirac(x) = 0

DiracDelta[x] == 0
Failure Successful Successful [Tested: 3] Successful [Tested: 1]
1.17.E2 ∫ - ∞ ∞ Ξ΄ ⁑ ( x - a ) ⁒ Ο• ⁒ ( x ) ⁒ d x = Ο• ⁒ ( a ) superscript subscript Dirac-delta π‘₯ π‘Ž italic-Ο• π‘₯ π‘₯ italic-Ο• π‘Ž {\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 lim n β†’ ∞ ⁑ n Ο€ ⁒ ∫ - ∞ ∞ e - n ⁒ ( x - a ) 2 ⁒ Ο• ⁒ ( x ) ⁒ d x = Ο• ⁒ ( a ) subscript β†’ 𝑛 𝑛 πœ‹ superscript subscript superscript 𝑒 𝑛 superscript π‘₯ π‘Ž 2 italic-Ο• π‘₯ π‘₯ italic-Ο• π‘Ž {\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 lim n β†’ ∞ ⁑ n Ο€ ⁒ ∫ - ∞ ∞ e - n ⁒ ( x - a ) 2 ⁒ Ο• ⁒ ( x ) ⁒ d x = 1 2 ⁒ Ο• ⁒ ( a - ) + 1 2 ⁒ Ο• ⁒ ( a + ) subscript β†’ 𝑛 𝑛 πœ‹ superscript subscript superscript 𝑒 𝑛 superscript π‘₯ π‘Ž 2 italic-Ο• π‘₯ π‘₯ 1 2 italic-Ο• limit-from π‘Ž 1 2 italic-Ο• limit-from π‘Ž {\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 1 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ e - i ⁒ a ⁒ t ⁒ ( ∫ - ∞ ∞ Ο• ⁒ ( x ) ⁒ e i ⁒ t ⁒ x ⁒ d x ) ⁒ d t = Ο• ⁒ ( a ) 1 2 πœ‹ superscript subscript superscript 𝑒 𝑖 π‘Ž 𝑑 superscript subscript italic-Ο• π‘₯ superscript 𝑒 𝑖 𝑑 π‘₯ π‘₯ 𝑑 italic-Ο• π‘Ž {\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 ∫ - ∞ ∞ ( 1 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ e i ⁒ ( x - a ) ⁒ t ⁒ d t ) ⁒ Ο• ⁒ ( x ) ⁒ d x = Ο• ⁒ ( a ) superscript subscript 1 2 πœ‹ superscript subscript superscript 𝑒 𝑖 π‘₯ π‘Ž 𝑑 𝑑 italic-Ο• π‘₯ π‘₯ italic-Ο• π‘Ž {\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 1 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ e - t 2 / ( 4 ⁒ n ) ⁒ e i ⁒ ( x - a ) ⁒ t ⁒ d t = n Ο€ ⁒ e - n ⁒ ( x - a ) 2 1 2 πœ‹ superscript subscript superscript 𝑒 superscript 𝑑 2 4 𝑛 superscript 𝑒 𝑖 π‘₯ π‘Ž 𝑑 𝑑 𝑛 πœ‹ superscript 𝑒 𝑛 superscript π‘₯ π‘Ž 2 {\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 Ξ΄ ⁑ ( x - a ) = 1 2 ⁒ Ο€ ⁒ ∫ - ∞ ∞ e i ⁒ ( x - a ) ⁒ t ⁒ d t Dirac-delta π‘₯ π‘Ž 1 2 πœ‹ superscript subscript superscript 𝑒 𝑖 π‘₯ π‘Ž 𝑑 𝑑 {\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 Ξ΄ ⁑ ( x - a ) = x ⁒ ∫ 0 ∞ t ⁒ J Ξ½ ⁑ ( x ⁒ t ) ⁒ J Ξ½ ⁑ ( a ⁒ t ) ⁒ d t Dirac-delta π‘₯ π‘Ž π‘₯ superscript subscript 0 𝑑 Bessel-J 𝜈 π‘₯ 𝑑 Bessel-J 𝜈 π‘Ž 𝑑 𝑑 {\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}		 β„œ ⁑ Ξ½ > - 1 , x > 0 , a > 0 formulae-sequence 𝜈 1 formulae-sequence π‘₯ 0 π‘Ž 0 {\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 Ξ΄ ⁑ ( x - a ) = 2 ⁒ x ⁒ a Ο€ ⁒ ∫ 0 ∞ t 2 ⁒ 𝗃 β„“ ⁑ ( x ⁒ t ) ⁒ 𝗃 β„“ ⁑ ( a ⁒ t ) ⁒ d t Dirac-delta π‘₯ π‘Ž 2 π‘₯ π‘Ž πœ‹ superscript subscript 0 superscript 𝑑 2 spherical-Bessel-J β„“ π‘₯ 𝑑 spherical-Bessel-J β„“ π‘Ž 𝑑 𝑑 {\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}		 x > 0 , a > 0 formulae-sequence π‘₯ 0 π‘Ž 0 {\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 Ξ΄ ⁑ ( x - a ) = ∫ - ∞ ∞ Ai ⁑ ( t - x ) ⁒ Ai ⁑ ( t - a ) ⁒ d t Dirac-delta π‘₯ π‘Ž superscript subscript Airy-Ai 𝑑 π‘₯ Airy-Ai 𝑑 π‘Ž 𝑑 {\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 1 2 ⁒ Ο€ ⁒ βˆ‘ k = - ∞ ∞ e - i ⁒ k ⁒ a ⁒ ( ∫ - Ο€ Ο€ Ο• ⁒ ( x ) ⁒ e i ⁒ k ⁒ x ⁒ d x ) = Ο• ⁒ ( a ) 1 2 πœ‹ superscript subscript π‘˜ superscript 𝑒 𝑖 π‘˜ π‘Ž superscript subscript πœ‹ πœ‹ italic-Ο• π‘₯ superscript 𝑒 𝑖 π‘˜ π‘₯ π‘₯ italic-Ο• π‘Ž {\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 ∫ - Ο€ Ο€ Ο• ⁒ ( x ) ⁒ ( 1 2 ⁒ Ο€ ⁒ βˆ‘ k = - ∞ ∞ e i ⁒ k ⁒ ( x - a ) ) ⁒ d x = Ο• ⁒ ( a ) superscript subscript πœ‹ πœ‹ italic-Ο• π‘₯ 1 2 πœ‹ superscript subscript π‘˜ superscript 𝑒 𝑖 π‘˜ π‘₯ π‘Ž π‘₯ italic-Ο• π‘Ž {\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 Ξ΄ ⁑ ( x - a ) = 1 2 ⁒ Ο€ ⁒ βˆ‘ k = - ∞ ∞ e i ⁒ k ⁒ ( x - a ) Dirac-delta π‘₯ π‘Ž 1 2 πœ‹ superscript subscript π‘˜ superscript 𝑒 𝑖 π‘˜ π‘₯ π‘Ž {\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 Ξ΄ ⁑ ( x - a ) = βˆ‘ k = 0 ∞ ( k + 1 2 ) ⁒ P k ⁑ ( x ) ⁒ P k ⁑ ( a ) Dirac-delta π‘₯ π‘Ž superscript subscript π‘˜ 0 π‘˜ 1 2 Legendre-spherical-polynomial π‘˜ π‘₯ Legendre-spherical-polynomial π‘˜ π‘Ž {\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 Ξ΄ ⁑ ( x - a ) = e - ( x + a ) / 2 ⁒ βˆ‘ k = 0 ∞ L k ⁑ ( x ) ⁒ L k ⁑ ( a ) Dirac-delta π‘₯ π‘Ž superscript 𝑒 π‘₯ π‘Ž 2 superscript subscript π‘˜ 0 shorthand-Laguerre-polynomial-L π‘˜ π‘₯ shorthand-Laguerre-polynomial-L π‘˜ π‘Ž {\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 Ξ΄ ⁑ ( x - a ) = e - ( x 2 + a 2 ) / 2 Ο€ ⁒ βˆ‘ k = 0 ∞ H k ⁑ ( x ) ⁒ H k ⁑ ( a ) 2 k ⁒ k ! Dirac-delta π‘₯ π‘Ž superscript 𝑒 superscript π‘₯ 2 superscript π‘Ž 2 2 πœ‹ superscript subscript π‘˜ 0 Hermite-polynomial-H π‘˜ π‘₯ Hermite-polynomial-H π‘˜ π‘Ž superscript 2 π‘˜ π‘˜ {\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 Ξ΄ ⁑ ( cos ⁑ ΞΈ 1 - cos ⁑ ΞΈ 2 ) ⁒ Ξ΄ ⁑ ( Ο• 1 - Ο• 2 ) = βˆ‘ β„“ = 0 ∞ βˆ‘ m = - β„“ β„“ Y β„“ , m ⁑ ( ΞΈ 1 , Ο• 1 ) ⁒ Y β„“ , m ⁑ ( ΞΈ 2 , Ο• 2 ) Β― Dirac-delta subscript πœƒ 1 subscript πœƒ 2 Dirac-delta subscript italic-Ο• 1 subscript italic-Ο• 2 superscript subscript β„“ 0 superscript subscript π‘š β„“ β„“ spherical-harmonic-Y β„“ π‘š subscript πœƒ 1 subscript italic-Ο• 1 spherical-harmonic-Y β„“ π‘š subscript πœƒ 2 subscript italic-Ο• 2 {\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
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