Coulomb Functions - 33.22 Particle Scattering and Atomic and Molecular Spectra

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
33.22.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{x}+\left({\sf k}^{2}-\frac{2Z}{x}-\frac{\ell(\ell+1)}{x^{2}}\right)w = 0}
\deriv[2]{w}{x}+\left({\sf k}^{2}-\frac{2Z}{x}-\frac{\ell(\ell+1)}{x^{2}}\right)w = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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]
Result: Complex[-0.5704416218017292, -1.0991449828236957]
Test Values: {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]}


Result: Complex[-2.110042339640732, -1.9880338717125847]
Test Values: {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]}

... skip entries to safe data
33.22#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r = -\eta\rho}
r = -\eta\rho		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
r = - eta*rho
 
r == - \[Eta]*\[Rho]
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \epsilon = 1/\eta^{2}}
\epsilon = 1/\eta^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
epsilon = 1/(eta)^(2)
 
\[Epsilon] == 1/\[Eta]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = 2\iunit\rho}
z = 2\iunit\rho		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
z = 2*I*rho

z == 2*I*\[Rho]
Failure Failure
Failed [70 / 70]
Result: 1.866025404-1.232050808*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .5000000000-.8660254040*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[1.8660254037844386, -1.2320508075688774]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.598076211353316, 1.4999999999999996]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
33.22#Ex13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \kappa = \iunit\eta}
\kappa = \iunit\eta		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
kappa = I*eta

\[Kappa] == I*\[Eta]
Failure Failure
Failed [96 / 100]
Result: 1.366025404-.3660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}


Result: 1.000000000-1.732050808*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [96 / 100]
Result: Complex[1.3660254037844386, -0.36602540378443876]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0, -1.7320508075688772]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
33.22#Ex14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \rho = z/(2\iunit)}
\rho = z/(2\iunit)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
rho = z/(2*I)

\[Rho] == z/(2*I)
Failure Failure
Failed [70 / 70]
Result: .6160254040+.9330127020*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .4330127020+.2500000000*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.6160254037844387, 0.9330127018922193]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7499999999999998, 1.299038105676658]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
33.22#Ex15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \eta = \kappa/\iunit}
\eta = \kappa/\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
eta = kappa/I

\[Eta] == \[Kappa]/I
Failure Failure
Failed [96 / 100]
Result: .3660254040+1.366025404*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2*3^(1/2)+1/2*I}


Result: 1.732050808+1.000000000*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, kappa = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [96 / 100]
Result: Complex[0.36602540378443876, 1.3660254037844386]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.7320508075688772, 1.0]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
33.22#Ex16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r = \kappa z/2}
r = \kappa z/2		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
r = kappa*z/2
 
r == \[Kappa]*z/2
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \epsilon = -1/\kappa^{2}}
\epsilon = -1/\kappa^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
epsilon = - 1/(kappa)^(2)
 
\[Epsilon] == - 1/\[Kappa]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \eta = +\epsilon^{-1/2}}
\eta = +\epsilon^{-1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
eta = + (epsilon)^(- 1/2)
 
\[Eta] == + \[Epsilon]^(- 1/2)
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \rho = -r/\eta}
\rho = -r/\eta		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
rho = - r/eta
 
\[Rho] == - r/\[Eta]
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex20 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \kappa = +(-\epsilon)^{-1/2}}
\kappa = +(-\epsilon)^{-1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
kappa = +(- epsilon)^(- 1/2)
 
\[Kappa] == +(- \[Epsilon])^(- 1/2)
 Skipped - no semantic math Skipped - no semantic math - -
33.22#Ex21 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = 2r/\kappa}
z = 2r/\kappa		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
z = 2*r/kappa
 
z == 2*r/\[Kappa]
 Skipped - no semantic math Skipped - no semantic math - -
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