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
Fail
-2.121320343-.7071067807*I <- {eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1}
-3.535533906+.7071067824*I <- {eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 2}
-5.656854249+2.828427125*I <- {eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 3}
-.7071067807+2.121320343*I <- {eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
Complex[-2.1213203435596424, -0.7071067811865475] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.5355339059327373, 0.7071067811865475] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5.656854249492381, 2.8284271247461903] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.949747468305832, -2.1213203435596424] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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 Error Successful -
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 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 Successful Successful
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
Fail
0.+3.999999998*I <- {eta = 2^(1/2)+I*2^(1/2), A[k] = 2^(1/2)+I*2^(1/2), ell = 1, k = 1}
-1.000000000+3.999999998*I <- {eta = 2^(1/2)+I*2^(1/2), A[k] = 2^(1/2)+I*2^(1/2), ell = 1, k = 2}
-.7071067810+3.292893217*I <- {eta = 2^(1/2)+I*2^(1/2), A[k] = 2^(1/2)+I*2^(1/2), ell = 1, k = 3}
-5.656854245+5.656854245*I <- {eta = 2^(1/2)+I*2^(1/2), A[k] = 2^(1/2)+I*2^(1/2), ell = 2, k = 1}
... skip entries to safe data
 Skip
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 Error Skip -
33.8#Ex1 a = 1 + ℓ + i ⁢ η 𝑎 1 ℓ imaginary-unit 𝜂 {\displaystyle{\displaystyle a=1+\ell+\mathrm{i}\eta}} a = 1 + ell + I*eta a = 1 + \[ScriptL]+ I*\[Eta] Failure Failure
Fail
.828427124+0.*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 1}
-.171572876+0.*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 2}
-1.171572876+0.*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 3}
-2.000000000+0.*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
0.8284271247461903 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-0.1715728752538097 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-1.1715728752538097 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-2.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.8#Ex1 a = 1 + ℓ - i ⁢ η 𝑎 1 ℓ imaginary-unit 𝜂 {\displaystyle{\displaystyle a=1+\ell-\mathrm{i}\eta}} a = 1 + ell - I*eta a = 1 + \[ScriptL]- I*\[Eta] Failure Failure
Fail
-2.000000000+2.828427124*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 1}
-3.000000000+2.828427124*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 2}
-4.000000000+2.828427124*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 3}
.828427124+2.828427124*I <- {a = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
Complex[-2.0, 2.8284271247461903] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.0, 2.8284271247461903] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-4.0, 2.8284271247461903] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8284271247461903, 2.8284271247461903] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.8#Ex2 b = - ℓ + i ⁢ η 𝑏 ℓ imaginary-unit 𝜂 {\displaystyle{\displaystyle b=-\ell+\mathrm{i}\eta}} b = - ell + I*eta b = - \[ScriptL]+ I*\[Eta] Failure Failure
Fail
3.828427124+0.*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 1}
4.828427124+0.*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 2}
5.828427124+0.*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 3}
1.000000000+0.*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
3.8284271247461903 <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
4.82842712474619 <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
5.82842712474619 <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
1.0 <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.8#Ex2 b = - ℓ - i ⁢ η 𝑏 ℓ imaginary-unit 𝜂 {\displaystyle{\displaystyle b=-\ell-\mathrm{i}\eta}} b = - ell - I*eta b = - \[ScriptL]- I*\[Eta] Failure Failure
Fail
1.000000000+2.828427124*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 1}
2.000000000+2.828427124*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 2}
3.000000000+2.828427124*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), ell = 3}
3.828427124+2.828427124*I <- {b = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
Complex[1.0, 2.8284271247461903] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.0, 2.8284271247461903] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.0, 2.8284271247461903] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.8284271247461903, 2.8284271247461903] <- {Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.8#Ex3 c = + i ⁢ ( 1 - ( η / ρ ) ) 𝑐 imaginary-unit 1 𝜂 𝜌 {\displaystyle{\displaystyle c=+\mathrm{i}(1-(\eta/\rho))}} c = + I*(1 -(eta/ rho)) c = + I*(1 -(\[Eta]/ \[Rho])) Failure Failure
Fail
1.414213562+1.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2)}
.414213562+.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)-I*2^(1/2)}
1.414213562-.585786438*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)-I*2^(1/2)}
2.414213562+.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, 0.41421356237309515] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -0.5857864376269049] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.414213562373095, 0.41421356237309515] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.8#Ex3 c = - i ⁢ ( 1 - ( η / ρ ) ) 𝑐 imaginary-unit 1 𝜂 𝜌 {\displaystyle{\displaystyle c=-\mathrm{i}(1-(\eta/\rho))}} c = - I*(1 -(eta/ rho)) c = - I*(1 -(\[Eta]/ \[Rho])) Failure Failure
Fail
1.414213562+1.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)+I*2^(1/2)}
2.414213562+2.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = 2^(1/2)-I*2^(1/2)}
1.414213562+3.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)-I*2^(1/2)}
.414213562+2.414213562*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), rho = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.414213562373095, 2.414213562373095] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, 3.414213562373095] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, 2.414213562373095] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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 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 Error Error - -
33.8#Ex5 F ℓ ′ = u ⁢ F ℓ diffop regular-Coulomb-F ℓ 1 𝑢 regular-Coulomb-F ℓ {\displaystyle{\displaystyle F_{\ell}'=uF_{\ell}}} subs( temp=u, diff( CoulombF(ell, =, temp), temp$(1) ) )*CoulombF(ell, $1, $2) Error Error Error - -
33.12.E8 d 2 w d z 2 = ( 4 ⁢ η 2 ⁢ ( 1 - z z ) + ℓ ⁢ ( ℓ + 1 ) z 2 ) ⁢ w derivative 𝑤 𝑧 2 4 superscript 𝜂 2 1 𝑧 𝑧 ℓ ℓ 1 superscript 𝑧 2 𝑤 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(% 4\eta^{2}\left(\frac{1-z}{z}\right)+\frac{\ell(\ell+1)}{z^{2}}\right)w}} diff(w, [z$(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
Fail
-23.33452376+7.334523764*I <- {eta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), ell = 1}
-24.74873732+8.748737326*I <- {eta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), ell = 2}
-26.87005766+10.87005767*I <- {eta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), ell = 3}
-5.920310202+21.92031019*I <- {eta = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
Complex[-23.334523779156072, 7.33452377915607] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-24.74873734152917, 8.748737341529166] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-26.87005768508881, 10.870057685088808] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[η, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[21.92031021678298, -5.920310216782976] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[η, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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
Fail
1.292893218+4.707106779*I <- {epsilon = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1}
-.1213203455+6.121320342*I <- {epsilon = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 2}
-2.242640688+8.242640685*I <- {epsilon = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 3}
4.707106779-1.292893218*I <- {epsilon = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), ell = 1}
... skip entries to safe data
Fail
Complex[1.2928932188134525, 4.707106781186548] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[ϵ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.12132034355964283, 6.121320343559644] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2], Rule[ϵ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.242640687119285, 8.242640687119286] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3], Rule[ϵ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.292893218813453, 0.7071067811865479] <- {Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1], Rule[ϵ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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 Error
33.14.E15 ∫ 0 ∞ ϕ n , ℓ 2 ⁢ ( r ) ⁢ d r = 1 superscript subscript 0 superscript subscript italic-ϕ 𝑛 ℓ 2 𝑟 𝑟 1 {\displaystyle{\displaystyle\int_{0}^{\infty}\phi_{n,\ell}^{2}(r)\mathrm{d}r=1}} int(phi(phi[n , ell])^(2)*(r), r = 0..infinity)= 1 Integrate[\[Phi](Subscript[\[Phi], n , \[ScriptL]])^(2)*(r), {r, 0, Infinity}]= 1 Failure Failure Skip
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[Integrate[Times[r, ϕ, Power[Subscript[ϕ, n, ℓ], 2]], {r, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[Integrate[Times[r, ϕ, Power[Subscript[ϕ, n, ℓ], 2]], {r, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[Integrate[Times[r, ϕ, Power[Subscript[ϕ, n, ℓ], 2]], {r, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[Integrate[Times[r, ϕ, Power[Subscript[ϕ, n, ℓ], 2]], {r, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
33.16#Ex1 ζ ℓ ⁢ ( ν , r ) = W ν , ℓ + 1 2 ⁡ ( 2 ⁢ r / ν ) subscript 𝜁 ℓ 𝜈 𝑟 Whittaker-confluent-hypergeometric-W 𝜈 ℓ 1 2 2 𝑟 𝜈 {\displaystyle{\displaystyle\zeta_{\ell}(\nu,r)=W_{\nu,\ell+\frac{1}{2}}\left(% 2r/\nu\right)}} zeta[ell]*(nu , r)= WhittakerW(nu, ell +(1)/(2), 2*r/ nu) Subscript[\[zeta], \[ScriptL]]*(\[Nu], r)= WhittakerW[\[Nu], \[ScriptL]+Divide[1,2], 2*r/ \[Nu]] Failure Failure Error Error
33.16#Ex2 ξ ℓ ⁢ ( ν , r ) = ℜ ⁡ ( e i ⁢ π ⁢ ν ⁢ W - ν , ℓ + 1 2 ⁡ ( e i ⁢ π ⁢ 2 ⁢ r / ν ) ) subscript 𝜉 ℓ 𝜈 𝑟 superscript 𝑒 imaginary-unit 𝜋 𝜈 Whittaker-confluent-hypergeometric-W 𝜈 ℓ 1 2 superscript 𝑒 imaginary-unit 𝜋 2 𝑟 𝜈 {\displaystyle{\displaystyle\xi_{\ell}(\nu,r)=\Re\left(e^{\mathrm{i}\pi\nu}W_{% -\nu,\ell+\frac{1}{2}}\left(e^{\mathrm{i}\pi}2r/\nu\right)\right)}} xi[ell]*(nu , r)= Re(exp(I*Pi*nu)*WhittakerW(- nu, ell +(1)/(2), exp(I*Pi)*2*r/ nu)) Subscript[\[Xi], \[ScriptL]]*(\[Nu], r)= Re[Exp[I*Pi*\[Nu]]*WhittakerW[- \[Nu], \[ScriptL]+Divide[1,2], Exp[I*Pi]*2*r/ \[Nu]]] Failure Failure Error Error
33.19#Ex4 δ 0 = ( β 2 ⁢ ℓ + 1 - 2 ⁢ ( ψ ⁡ ( 2 ⁢ ℓ + 2 ) + ψ ⁡ ( 1 ) ) ⁢ A ⁢ ( ϵ , ℓ ) ) ⁢ α 0 subscript 𝛿 0 subscript 𝛽 2 ℓ 1 2 digamma 2 ℓ 2 digamma 1 𝐴 italic-ϵ ℓ subscript 𝛼 0 {\displaystyle{\displaystyle\delta_{0}=\left(\beta_{2\ell+1}-2(\psi\left(2\ell% +2\right)+\psi\left(1\right))A(\epsilon,\ell)\right)\alpha_{0}}} delta[0]=(beta[2*ell + 1]- 2*(Psi(2*ell + 2)+ Psi(1))*A*(epsilon , ell))* alpha[0] Subscript[\[Delta], 0]=(Subscript[\[Beta], 2*\[ScriptL]+ 1]- 2*(PolyGamma[2*\[ScriptL]+ 2]+ PolyGamma[1])*A*(\[Epsilon], \[ScriptL]))* Subscript[\[Alpha], 0] Failure Failure Skip Error
33.19#Ex5 δ 1 = ( β 2 ⁢ ℓ + 2 - 2 ⁢ ( ψ ⁡ ( 2 ⁢ ℓ + 3 ) + ψ ⁡ ( 2 ) ) ⁢ A ⁢ ( ϵ , ℓ ) ) ⁢ α 1 subscript 𝛿 1 subscript 𝛽 2 ℓ 2 2 digamma 2 ℓ 3 digamma 2 𝐴 italic-ϵ ℓ subscript 𝛼 1 {\displaystyle{\displaystyle\delta_{1}=\left(\beta_{2\ell+2}-2(\psi\left(2\ell% +3\right)+\psi\left(2\right))A(\epsilon,\ell)\right)\alpha_{1}}} delta[1]=(beta[2*ell + 2]- 2*(Psi(2*ell + 3)+ Psi(2))*A*(epsilon , ell))* alpha[1] Subscript[\[Delta], 1]=(Subscript[\[Beta], 2*\[ScriptL]+ 2]- 2*(PolyGamma[2*\[ScriptL]+ 3]+ PolyGamma[2])*A*(\[Epsilon], \[ScriptL]))* Subscript[\[Alpha], 1] Failure Failure Skip Error
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
Fail
-1.414213562-9.414213558*I <- {Z = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1, k = 1, x = 1}
.7071067810-3.292893217*I <- {Z = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1, k = 1, x = 2}
1.099943881-1.566722784*I <- {Z = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1, k = 1, x = 3}
2.828427124-5.171572872*I <- {Z = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), ell = 1, k = 2, x = 1}
... skip entries to safe data
Fail
Complex[-1.414213562373095, -9.414213562373096] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1]}
Complex[-7.071067811865475, -15.071067811865476] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 2]}
Complex[-15.556349186104047, -23.556349186104047] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 3]}
Complex[0.7071067811865477, -3.292893218813453] <- {Rule[k, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ℓ, 1]}
... skip entries to safe data
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
Fail
4.242640686-1.414213562*I <- {rho = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
4.242640686-4.242640686*I <- {rho = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.414213562-4.242640686*I <- {rho = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.414213562-1.414213562*I <- {rho = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[4.242640687119286, -1.4142135623730951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 4.242640687119286] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[4.242640687119286, 4.242640687119286] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.22#Ex13 κ = i ⁢ η 𝜅 imaginary-unit 𝜂 {\displaystyle{\displaystyle\kappa=\mathrm{i}\eta}} kappa = I*eta \[Kappa]= I*\[Eta] Failure Failure Skip Successful
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
Fail
.7071067810+2.121320343*I <- {rho = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
2.121320343+2.121320343*I <- {rho = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
2.121320343+.7071067810*I <- {rho = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.7071067810+.7071067810*I <- {rho = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.7071067811865477, 2.121320343559643] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7071067811865477, -0.7071067811865477] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.121320343559643, -0.7071067811865477] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.121320343559643, 2.121320343559643] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ρ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
33.22#Ex15 η = κ / i 𝜂 𝜅 imaginary-unit {\displaystyle{\displaystyle\eta=\kappa/\mathrm{i}}} eta = kappa/ I \[Eta]= \[Kappa]/ I Failure Failure Skip Successful
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