Results of Coulomb Functions

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
33.2.E1 d 2 w d ρ 2 + ( 1 - 2 ⁢ η ρ - ℓ ⁢ ( ℓ + 1 ) ρ 2 ) ⁢ w = 0 derivative 𝑤 𝜌 2 1 2 𝜂 𝜌 ℓ ℓ 1 superscript 𝜌 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\rho}^{2}}+% \left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^{2}}\right)w=0}} diff(w, [rho$(2)])+(1 -(2*eta)/(rho)-(ell*(ell + 1))/((rho)^(2)))* w = 0 D[w, {\[Rho], 2}]+(1 -Divide[2*\[Eta],\[Rho]]-Divide[\[ScriptL]*(\[ScriptL]+ 1),\[Rho]^(2)])* w == 0 Failure Failure
Failed [300 / 300]
300/300]: [[-11.25833025+5.499999998*I <- {eta = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, ell = 3}
-5.499999998-11.25833025*I <- {eta = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2), ell = 3}
Failed [294 / 300]
{Complex[-11.258330249197703, 5.5] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 3], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[10.2583302491977, -3.767949192431125] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 3], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
33.4.E2 R ℓ ⁢ X ℓ - 1 - T ℓ ⁢ X ℓ + R ℓ + 1 ⁢ X ℓ + 1 = 0 subscript 𝑅 ℓ subscript 𝑋 ℓ 1 subscript 𝑇 ℓ subscript 𝑋 ℓ subscript 𝑅 ℓ 1 subscript 𝑋 ℓ 1 0 {\displaystyle{\displaystyle R_{\ell}X_{\ell-1}-T_{\ell}X_{\ell}+R_{\ell+1}X_{% \ell+1}=0}} R[ell]*X[ell - 1]- T[ell]*X[ell]+ R[ell + 1]*X[ell + 1] = 0 Subscript[R, \[ScriptL]]*Subscript[X, \[ScriptL]- 1]- Subscript[T, \[ScriptL]]*Subscript[X, \[ScriptL]]+ Subscript[R, \[ScriptL]+ 1]*Subscript[X, \[ScriptL]+ 1] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.5#Ex7 F ℓ ⁡ ( 0 , ρ ) = ( π ⁢ ρ / 2 ) 1 / 2 ⁢ J ℓ + 1 2 ⁡ ( ρ ) regular-Coulomb-F ℓ 0 𝜌 superscript 𝜋 𝜌 2 1 2 Bessel-J ℓ 1 2 𝜌 {\displaystyle{\displaystyle F_{\ell}\left(0,\rho\right)=(\pi\rho/2)^{1/2}J_{% \ell+\frac{1}{2}}\left(\rho\right)}} CoulombF(ell, 0, rho) = (Pi*rho/ 2)^(1/ 2)* BesselJ(ell +(1)/(2), rho) Error Failure Missing Macro Error Error -
33.5#Ex9 F 0 ⁡ ( 0 , ρ ) = sin ⁡ ρ regular-Coulomb-F 0 0 𝜌 𝜌 {\displaystyle{\displaystyle F_{0}\left(0,\rho\right)=\sin\rho}} CoulombF(0, 0, rho) = sin(rho) Error Successful Missing Macro Error - -
33.5.E6 2 ℓ ⁢ ℓ ! ( 2 ⁢ ℓ + 1 ) ! = 1 ( 2 ⁢ ℓ + 1 ) !! superscript 2 ℓ ℓ 2 ℓ 1 1 double-factorial 2 ℓ 1 {\displaystyle{\displaystyle\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1% )!!}}} ((2)^(ell)* factorial(ell))/(factorial(2*ell + 1)) = (1)/(doublefactorial(2*ell + 1)) Divide[(2)^\[ScriptL]* (\[ScriptL])!,(2*\[ScriptL]+ 1)!] == Divide[1,(2*\[ScriptL]+ 1)!!] Failure Failure Error
Failed [1 / 1]
{Plus[Times[Power[2.0, ℓ], Factorial[ℓ], Power[Factorial[Plus[1.0, Times[2.0, ℓ]]], -1]], Times[-1.0, Power[Factorial2[Plus[1.0, Times[2.0, ℓ]]], -1]]] <- {}
33.6.E3 ( k + ℓ ) ⁢ ( k - ℓ - 1 ) ⁢ A k ℓ = 2 ⁢ η ⁢ A k - 1 ℓ - A k - 2 ℓ 𝑘 ℓ 𝑘 ℓ 1 superscript subscript 𝐴 𝑘 ℓ 2 𝜂 superscript subscript 𝐴 𝑘 1 ℓ superscript subscript 𝐴 𝑘 2 ℓ {\displaystyle{\displaystyle(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{k-1}^{\ell% }-A_{k-2}^{\ell}}} (k + ell)*(k - ell - 1)* (A[k])^(ell) = 2*eta*(A[k - 1])^(ell)- (A[k - 2])^(ell) (k + \[ScriptL])*(k - \[ScriptL]- 1)* (Subscript[A, k])^\[ScriptL] == 2*\[Eta]*(Subscript[A, k - 1])^\[ScriptL]- (Subscript[A, k - 2])^\[ScriptL] Skipped - no semantic math Skipped - no semantic math - -
33.6.E4 A k ℓ ⁢ ( η ) = ( - i ) k - ℓ - 1 ( k - ℓ - 1 ) ! ⁢ F 1 2 ⁡ ( ℓ + 1 - k , ℓ + 1 - i ⁢ η ; 2 ⁢ ℓ + 2 ; 2 ) superscript subscript 𝐴 𝑘 ℓ 𝜂 superscript imaginary-unit 𝑘 ℓ 1 𝑘 ℓ 1 Gauss-hypergeometric-F-as-2F1 ℓ 1 𝑘 ℓ 1 imaginary-unit 𝜂 2 ℓ 2 2 {\displaystyle{\displaystyle A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1% }}{(k-\ell-1)!}\*{{}_{2}F_{1}}\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2% \right)}} (A[k])^(ell)*(eta) = ((- I)^(k - ell - 1))/(factorial(k - ell - 1))* hypergeom([ell + 1 - k , ell + 1 - I*eta], [2*ell + 2], 2) (Subscript[A, k])^\[ScriptL]*(\[Eta]) == Divide[(- I)^(k - \[ScriptL]- 1),(k - \[ScriptL]- 1)!]* HypergeometricPFQ[{\[ScriptL]+ 1 - k , \[ScriptL]+ 1 - I*\[Eta]}, {2*\[ScriptL]+ 2}, 2] Failure Failure Error
Failed [293 / 300]
{Complex[0.5000000000000001, 0.8660254037844386] <- {Rule[k, 1], Rule[ℓ, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[0.0, 1.0] <- {Rule[k, 1], Rule[ℓ, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
33.7.E1 F ℓ ⁡ ( η , ρ ) = ρ ℓ + 1 ⁢ 2 ℓ ⁢ e i ⁢ ρ - ( π ⁢ η / 2 ) | Γ ⁡ ( ℓ + 1 + i ⁢ η ) | ⁢ ∫ 0 1 e - 2 ⁢ i ⁢ ρ ⁢ t ⁢ t ℓ + i ⁢ η ⁢ ( 1 - t ) ℓ - i ⁢ η ⁢ d t regular-Coulomb-F ℓ 𝜂 𝜌 superscript 𝜌 ℓ 1 superscript 2 ℓ superscript 𝑒 imaginary-unit 𝜌 𝜋 𝜂 2 Euler-Gamma ℓ 1 imaginary-unit 𝜂 superscript subscript 0 1 superscript 𝑒 2 imaginary-unit 𝜌 𝑡 superscript 𝑡 ℓ imaginary-unit 𝜂 superscript 1 𝑡 ℓ imaginary-unit 𝜂 𝑡 {\displaystyle{\displaystyle F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1% }2^{\ell}e^{\mathrm{i}\rho-(\pi\eta/2)}}{|\Gamma\left(\ell+1+\mathrm{i}\eta% \right)|}\int_{0}^{1}e^{-2\mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-% \mathrm{i}\eta}\mathrm{d}t}} CoulombF(ell, eta, rho) = ((rho)^(ell + 1)* (2)^(ell)* exp(I*rho -(Pi*eta/ 2)))/(abs(GAMMA(ell + 1 + I*eta)))*int(exp(- 2*I*rho*t)*(t)^(ell + I*eta)*(1 - t)^(ell - I*eta), t = 0..1) Error Failure Missing Macro Error Error -
33.8#Ex4 F ℓ = + ( q - 1 ⁢ ( u - p ) 2 + q ) - 1 / 2 regular-Coulomb-F ℓ superscript superscript 𝑞 1 superscript 𝑢 𝑝 2 𝑞 1 2 {\displaystyle{\displaystyle F_{\ell}=+(q^{-1}(u-p)^{2}+q)^{-1/2}}} CoulombF(ell, =, +)*((q)^(- 1)*(u - p)^(2)+ q)^(- 1/ 2) Error Translation Error Missing Macro Error - -
33.8#Ex4 F ℓ = - ( q - 1 ⁢ ( u - p ) 2 + q ) - 1 / 2 regular-Coulomb-F ℓ superscript superscript 𝑞 1 superscript 𝑢 𝑝 2 𝑞 1 2 {\displaystyle{\displaystyle F_{\ell}=-(q^{-1}(u-p)^{2}+q)^{-1/2}}} CoulombF(ell, =, -)*((q)^(- 1)*(u - p)^(2)+ q)^(- 1/ 2) Error Translation Error Missing Macro Error - -
33.9.E2 k ⁢ ( k + 2 ⁢ ℓ + 1 ) 2 ⁢ k + 2 ⁢ ℓ + 1 ⁢ a k - 2 ⁢ η ⁢ a k - 1 + ( k - 2 ) ⁢ ( k + 2 ⁢ ℓ - 1 ) 2 ⁢ k + 2 ⁢ ℓ - 3 ⁢ a k - 2 = 0 𝑘 𝑘 2 ℓ 1 2 𝑘 2 ℓ 1 subscript 𝑎 𝑘 2 𝜂 subscript 𝑎 𝑘 1 𝑘 2 𝑘 2 ℓ 1 2 𝑘 2 ℓ 3 subscript 𝑎 𝑘 2 0 {\displaystyle{\displaystyle\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{k-1}% +\frac{(k-2)(k+2\ell-1)}{2k+2\ell-3}a_{k-2}=0}} (k*(k + 2*ell + 1))/(2*k + 2*ell + 1)*a[k]- 2*eta*a[k - 1]+((k - 2)*(k + 2*ell - 1))/(2*k + 2*ell - 3)*a[k - 2] = 0 Divide[k*(k + 2*\[ScriptL]+ 1),2*k + 2*\[ScriptL]+ 1]*Subscript[a, k]- 2*\[Eta]*Subscript[a, k - 1]+Divide[(k - 2)*(k + 2*\[ScriptL]- 1),2*k + 2*\[ScriptL]- 3]*Subscript[a, k - 2] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.9.E5 4 ⁢ η 2 ⁢ ( k - 2 ⁢ ℓ ) ⁢ b k + 1 + k ⁢ b k - 1 + b k - 2 = 0 4 superscript 𝜂 2 𝑘 2 ℓ subscript 𝑏 𝑘 1 𝑘 subscript 𝑏 𝑘 1 subscript 𝑏 𝑘 2 0 {\displaystyle{\displaystyle 4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2}=0}} 4*(eta)^(2)*(k - 2*ell)* b[k + 1]+ k*b[k - 1]+ b[k - 2] = 0 4*\[Eta]^(2)*(k - 2*\[ScriptL])* Subscript[b, k + 1]+ k*Subscript[b, k - 1]+ Subscript[b, k - 2] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.12#Ex6 B 1 = - 1 5 ⁢ x subscript 𝐵 1 1 5 𝑥 {\displaystyle{\displaystyle B_{1}=-\tfrac{1}{5}x}} B[1] = -(1)/(5)*x Subscript[B, 1] == -Divide[1,5]*x Skipped - no semantic math Skipped - no semantic math - -
33.12#Ex7 B 2 = 1 350 ⁢ ( 7 ⁢ x 5 - 30 ⁢ x 2 ) subscript 𝐵 2 1 350 7 superscript 𝑥 5 30 superscript 𝑥 2 {\displaystyle{\displaystyle B_{2}=\tfrac{1}{350}(7x^{5}-30x^{2})}} B[2] = (1)/(350)*(7*(x)^(5)- 30*(x)^(2)) Subscript[B, 2] == Divide[1,350]*(7*(x)^(5)- 30*(x)^(2)) Skipped - no semantic math Skipped - no semantic math - -
33.12#Ex8 B 3 = 1 15750 ⁢ ( 264 ⁢ x 6 - 290 ⁢ x 3 - 560 ) subscript 𝐵 3 1 15750 264 superscript 𝑥 6 290 superscript 𝑥 3 560 {\displaystyle{\displaystyle B_{3}=\tfrac{1}{15750}(264x^{6}-290x^{3}-560)}} B[3] = (1)/(15750)*(264*(x)^(6)- 290*(x)^(3)- 560) Subscript[B, 3] == Divide[1,15750]*(264*(x)^(6)- 290*(x)^(3)- 560) Skipped - no semantic math Skipped - no semantic math - -
33.12.E8 d 2 w d z 2 = ( 4 ⁢ η 2 ⁢ ( 1 - z z ) + ℓ ⁢ ( ℓ + 1 ) z 2 ) ⁢ w derivative 𝑤 𝑧 2 4 superscript 𝜂 2 1 𝑧 𝑧 ℓ ℓ 1 superscript 𝑧 2 𝑤 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(% 4\eta^{2}\left(\frac{1-z}{z}\right)+\frac{\ell(\ell+1)}{z^{2}}\right)w}} diff(w, [z$(2)]) = (4*(eta)^(2)*((1 - z)/(z))+(ell*(ell + 1))/((z)^(2)))* w D[w, {z, 2}] == (4*\[Eta]^(2)*(Divide[1 - z,z])+Divide[\[ScriptL]*(\[ScriptL]+ 1),(z)^(2)])* w Failure Failure Error
Failed [296 / 300]
{Complex[-3.7320508075688767, 1.5358983848622458] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[-7.196152422706632, 3.535898384862246] <- {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
33.14.E1 d 2 w d r 2 + ( ϵ + 2 r - ℓ ⁢ ( ℓ + 1 ) r 2 ) ⁢ w = 0 derivative 𝑤 𝑟 2 italic-ϵ 2 𝑟 ℓ ℓ 1 superscript 𝑟 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}r}^{2}}+\left(% \epsilon+\frac{2}{r}-\frac{\ell(\ell+1)}{r^{2}}\right)w=0}} diff(w, [r$(2)])+(epsilon +(2)/(r)-(ell*(ell + 1))/((r)^(2)))* w = 0 D[w, {r, 2}]+(\[Epsilon]+Divide[2,r]-Divide[\[ScriptL]*(\[ScriptL]+ 1),(r)^(2)])* w == 0 Failure Failure Error
Failed [300 / 300]
{Complex[-1.0584754935143141, -0.611111111111111] <- {Rule[r, Rational[-3, 2]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[ϵ, 1]}
Complex[-0.19245008972987526, -0.11111111111111109] <- {Rule[r, Rational[-3, 2]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1], Rule[ϵ, 2]}
33.14#Ex1 r = - η ⁢ ρ 𝑟 𝜂 𝜌 {\displaystyle{\displaystyle r=-\eta\rho}} r = - eta*rho r == - \[Eta]*\[Rho] Skipped - no semantic math Skipped - no semantic math - -
33.14#Ex2 ϵ = 1 / η 2 italic-ϵ 1 superscript 𝜂 2 {\displaystyle{\displaystyle\epsilon=1/\eta^{2}}} epsilon = 1/ (eta)^(2) \[Epsilon] == 1/ \[Eta]^(2) Skipped - no semantic math Skipped - no semantic math - -
33.14.E12 A ⁢ ( ϵ , ℓ ) = Γ ⁡ ( 1 + ℓ + κ ) Γ ⁡ ( κ - ℓ ) ⁢ κ - 2 ⁢ ℓ - 1 𝐴 italic-ϵ ℓ Euler-Gamma 1 ℓ 𝜅 Euler-Gamma 𝜅 ℓ superscript 𝜅 2 ℓ 1 {\displaystyle{\displaystyle A(\epsilon,\ell)=\frac{\Gamma\left(1+\ell+\kappa% \right)}{\Gamma\left(\kappa-\ell\right)}\kappa^{-2\ell-1}}} A*(epsilon , ell) = (GAMMA(1 + ell + kappa))/(GAMMA(kappa - ell))*(kappa)^(- 2*ell - 1) A*(\[Epsilon], \[ScriptL]) == Divide[Gamma[1 + \[ScriptL]+ \[Kappa]],Gamma[\[Kappa]- \[ScriptL]]]*\[Kappa]^(- 2*\[ScriptL]- 1) Failure Failure Error
Failed [6 / 6]
{1.4444444444444444 <- {Rule[ℓ, 1], Rule[ϵ, 1], Rule[κ, Rational[3, 2]]}
Complex[2.4444444444444446, 0.0] <- {Rule[ℓ, 1], Rule[ϵ, 2], Rule[κ, Rational[3, 2]]}
33.14.E15 ∫ 0 ∞ ϕ m , ℓ ⁢ ( r ) ⁢ ϕ n , ℓ ⁢ ( r ) ⁢ d r = δ m , n superscript subscript 0 subscript italic-ϕ 𝑚 ℓ 𝑟 subscript italic-ϕ 𝑛 ℓ 𝑟 𝑟 Kronecker 𝑚 𝑛 {\displaystyle{\displaystyle\int_{0}^{\infty}\phi_{m,\ell}(r)\phi_{n,\ell}(r)% \mathrm{d}r=\delta_{m,n}}} int(phi[m , ell]*(r)* phi[n , ell]*(r), r = 0..infinity) = KroneckerDelta[m, n] Integrate[Subscript[\[Phi], m , \[ScriptL]]*(r)* Subscript[\[Phi], n , \[ScriptL]]*(r), {r, 0, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n] Translation Error Translation Error - -
33.19.E4 γ k - γ k - 1 + 1 4 ⁢ ( k - 1 ) ⁢ ( k - 2 ⁢ ℓ - 2 ) ⁢ ϵ ⁢ γ k - 2 = 0 subscript 𝛾 𝑘 subscript 𝛾 𝑘 1 1 4 𝑘 1 𝑘 2 ℓ 2 italic-ϵ subscript 𝛾 𝑘 2 0 {\displaystyle{\displaystyle\gamma_{k}-\gamma_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-% 2)\epsilon\gamma_{k-2}=0}} gamma[k]- gamma[k - 1]+(1)/(4)*(k - 1)*(k - 2*ell - 2)* epsilon*gamma[k - 2] = 0 Subscript[\[Gamma], k]- Subscript[\[Gamma], k - 1]+Divide[1,4]*(k - 1)*(k - 2*\[ScriptL]- 2)* \[Epsilon]*Subscript[\[Gamma], k - 2] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.19.E6 k ⁢ ( k + 2 ⁢ ℓ + 1 ) ⁢ δ k + 2 ⁢ δ k - 1 + ϵ ⁢ δ k - 2 + 2 ⁢ ( 2 ⁢ k + 2 ⁢ ℓ + 1 ) ⁢ A ⁢ ( ϵ , ℓ ) ⁢ α k = 0 𝑘 𝑘 2 ℓ 1 subscript 𝛿 𝑘 2 subscript 𝛿 𝑘 1 italic-ϵ subscript 𝛿 𝑘 2 2 2 𝑘 2 ℓ 1 𝐴 italic-ϵ ℓ subscript 𝛼 𝑘 0 {\displaystyle{\displaystyle k(k+2\ell+1)\delta_{k}+2\delta_{k-1}+\epsilon% \delta_{k-2}+2(2k+2\ell+1)A(\epsilon,\ell)\alpha_{k}=0}} k*(k + 2*ell + 1)* delta[k]+ 2*delta[k - 1]+ epsilon*delta[k - 2]+ 2*(2*k + 2*ell + 1)* A*(epsilon , ell)* alpha[k] = 0 k*(k + 2*\[ScriptL]+ 1)* Subscript[\[Delta], k]+ 2*Subscript[\[Delta], k - 1]+ \[Epsilon]*Subscript[\[Delta], k - 2]+ 2*(2*k + 2*\[ScriptL]+ 1)* A*(\[Epsilon], \[ScriptL])* Subscript[\[Alpha], k] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.19.E7 β k - β k - 1 + 1 4 ⁢ ( k - 1 ) ⁢ ( k - 2 ⁢ ℓ - 2 ) ⁢ ϵ ⁢ β k - 2 + 1 2 ⁢ ( k - 1 ) ⁢ ϵ ⁢ γ k - 2 = 0 subscript 𝛽 𝑘 subscript 𝛽 𝑘 1 1 4 𝑘 1 𝑘 2 ℓ 2 italic-ϵ subscript 𝛽 𝑘 2 1 2 𝑘 1 italic-ϵ subscript 𝛾 𝑘 2 0 {\displaystyle{\displaystyle\beta_{k}-\beta_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)% \epsilon\beta_{k-2}+\tfrac{1}{2}(k-1)\epsilon\gamma_{k-2}=0}} beta[k]- beta[k - 1]+(1)/(4)*(k - 1)*(k - 2*ell - 2)* epsilon*beta[k - 2]+(1)/(2)*(k - 1)* epsilon*gamma[k - 2] = 0 Subscript[\[Beta], k]- Subscript[\[Beta], k - 1]+Divide[1,4]*(k - 1)*(k - 2*\[ScriptL]- 2)* \[Epsilon]*Subscript[\[Beta], k - 2]+Divide[1,2]*(k - 1)* \[Epsilon]*Subscript[\[Gamma], k - 2] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.20#Ex5 C k , p = 0 subscript 𝐶 𝑘 𝑝 0 {\displaystyle{\displaystyle C_{k,p}=0}} C[k , p] = 0 Subscript[C, k , p] == 0 Skipped - no semantic math Skipped - no semantic math - -
33.20#Ex6 C k , p = ( - ( 2 ⁢ ℓ + p ) ⁢ C k - 1 , p - 2 + C k - 1 , p - 3 ) / ( 4 ⁢ p ) subscript 𝐶 𝑘 𝑝 2 ℓ 𝑝 subscript 𝐶 𝑘 1 𝑝 2 subscript 𝐶 𝑘 1 𝑝 3 4 𝑝 {\displaystyle{\displaystyle C_{k,p}=\left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}% \right)/(4p)}} C[k , p] = (-(2*ell + p)*C[k - 1 , p - 2]+ C[k - 1 , p - 3])/(4*p) Subscript[C, k , p] == (-(2*\[ScriptL]+ p)*Subscript[C, k - 1 , p - 2]+ Subscript[C, k - 1 , p - 3])/(4*p) Skipped - no semantic math Skipped - no semantic math - -
33.22.E3 d 2 w d x 2 + ( 𝗄 2 - 2 ⁢ Z x - ℓ ⁢ ( ℓ + 1 ) x 2 ) ⁢ w = 0 derivative 𝑤 𝑥 2 superscript 𝗄 2 2 𝑍 𝑥 ℓ ℓ 1 superscript 𝑥 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+\left(% {\sf k}^{2}-\frac{2Z}{x}-\frac{\ell(\ell+1)}{x^{2}}\right)w=0}} diff(w, [x$(2)])+((k)^(2)-(2*Z)/(x)-(ell*(ell + 1))/((x)^(2)))* w = 0 D[w, {x, 2}]+((k)^(2)-Divide[2*Z,x]-Divide[\[ScriptL]*(\[ScriptL]+ 1),(x)^(2)])* w == 0 Failure Failure Error
Failed [297 / 300]
{Complex[-0.5704416218017292, -1.0991449828236957] <- {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 1]}
Complex[-2.110042339640732, -1.9880338717125847] <- {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, Rational[3, 2]], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ℓ, 2]}
33.22#Ex10 r = - η ⁢ ρ 𝑟 𝜂 𝜌 {\displaystyle{\displaystyle r=-\eta\rho}} r = - eta*rho r == - \[Eta]*\[Rho] Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex11 ϵ = 1 / η 2 italic-ϵ 1 superscript 𝜂 2 {\displaystyle{\displaystyle\epsilon=1/\eta^{2}}} epsilon = 1/ (eta)^(2) \[Epsilon] == 1/ \[Eta]^(2) Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex12 z = 2 ⁢ i ⁢ ρ 𝑧 2 imaginary-unit 𝜌 {\displaystyle{\displaystyle z=2\mathrm{i}\rho}} z = 2*I*rho z == 2*I*\[Rho] Failure Failure
Failed [70 / 70]
70/70]: [[1.866025404-1.232050808*I <- {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}
.5000000000-.8660254040*I <- {rho = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}
Failed [70 / 70]
{Complex[1.8660254037844386, -1.2320508075688774] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[2.598076211353316, 1.4999999999999996] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
33.22#Ex13 κ = i ⁢ η 𝜅 imaginary-unit 𝜂 {\displaystyle{\displaystyle\kappa=\mathrm{i}\eta}} kappa = I*eta \[Kappa] == I*\[Eta] Failure Failure
Failed [96 / 100]
96/100]: [[1.366025404-.3660254040*I <- {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}
1.000000000-1.732050808*I <- {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2-1/2*I*3^(1/2)}
Failed [96 / 100]
{Complex[1.3660254037844386, -0.36602540378443876] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[1.0, -1.7320508075688772] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}
33.22#Ex14 ρ = z / ( 2 ⁢ i ) 𝜌 𝑧 2 imaginary-unit {\displaystyle{\displaystyle\rho=z/(2\mathrm{i})}} rho = z/(2*I) \[Rho] == z/(2*I) Failure Failure
Failed [70 / 70]
70/70]: [[.6160254040+.9330127020*I <- {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}
.4330127020+.2500000000*I <- {rho = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}
Failed [70 / 70]
{Complex[0.6160254037844387, 0.9330127018922193] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[-0.7499999999999998, 1.299038105676658] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
33.22#Ex15 η = κ / i 𝜂 𝜅 imaginary-unit {\displaystyle{\displaystyle\eta=\kappa/\mathrm{i}}} eta = kappa/ I \[Eta] == \[Kappa]/ I Failure Failure
Failed [96 / 100]
96/100]: [[.3660254040+1.366025404*I <- {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}
1.732050808+1.000000000*I <- {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2-1/2*I*3^(1/2)}
Failed [96 / 100]
{Complex[0.36602540378443876, 1.3660254037844386] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Complex[1.7320508075688772, 1.0] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}
33.22#Ex16 r = κ ⁢ z / 2 𝑟 𝜅 𝑧 2 {\displaystyle{\displaystyle r=\kappa z/2}} r = kappa*z/ 2 r == \[Kappa]*z/ 2 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex17 ϵ = - 1 / κ 2 italic-ϵ 1 superscript 𝜅 2 {\displaystyle{\displaystyle\epsilon=-1/\kappa^{2}}} epsilon = - 1/ (kappa)^(2) \[Epsilon] == - 1/ \[Kappa]^(2) Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex18 η = + ϵ - 1 / 2 𝜂 superscript italic-ϵ 1 2 {\displaystyle{\displaystyle\eta=+\epsilon^{-1/2}}} eta = + (epsilon)^(- 1/ 2) \[Eta] == + \[Epsilon]^(- 1/ 2) Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex19 ρ = - r / η 𝜌 𝑟 𝜂 {\displaystyle{\displaystyle\rho=-r/\eta}} rho = - r/ eta \[Rho] == - r/ \[Eta] Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex20 κ = + ( - ϵ ) - 1 / 2 𝜅 superscript italic-ϵ 1 2 {\displaystyle{\displaystyle\kappa=+(-\epsilon)^{-1/2}}} kappa = +(- epsilon)^(- 1/ 2) \[Kappa] == +(- \[Epsilon])^(- 1/ 2) Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex21 z = 2 ⁢ r / κ 𝑧 2 𝑟 𝜅 {\displaystyle{\displaystyle z=2r/\kappa}} z = 2*r/ kappa z == 2*r/ \[Kappa] Skipped - no semantic math Skipped - no semantic math - -
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