Expression 10: "Y" equals StartRoot, StartFraction, left parenthesis, 2 "l" plus 1 , right parenthesis Over 4 pi , EndFraction StartFraction, left parenthesis, "l" minus "m" , right parenthesis ! Over left parenthesis, "l" plus "m" , right parenthesis ! , EndFraction , EndRoot "P" Subscript, "l" "m" , Baseline left parenthesis, cos phi , right parenthesis left parenthesis, negative 1 , right parenthesis Superscript, StartFraction, "m" psi Over pi , EndFraction , BaselineY=2l+14πl−m!l+m!Plmcosϕ−1mψπ
equals=
0.6 3 0 7 8 3 1 3 0 5 0 50.630783130505
10
Expression 11: "Y" Subscript, "R" "e" , Baseline equals StartRoot, StartFraction, left parenthesis, 2 "l" plus 1 , right parenthesis Over 4 pi , EndFraction StartFraction, left parenthesis, "l" minus "m" , right parenthesis ! Over left parenthesis, "l" plus "m" , right parenthesis ! , EndFraction , EndRoot "P" Subscript, "l" "m" , Baseline left parenthesis, cos phi , right parenthesis cosine left parenthesis, "m" psi , right parenthesisYRe=2l+14πl−m!l+m!Plmcosϕcosmψ
equals=
0.6 3 0 7 8 3 1 3 0 5 0 50.630783130505
11
Expression 12: "Y" Subscript, "I" "m" , Baseline equals StartRoot, StartFraction, left parenthesis, 2 "l" plus 1 , right parenthesis Over 4 pi , EndFraction StartFraction, left parenthesis, "l" minus "m" , right parenthesis ! Over left parenthesis, "l" plus "m" , right parenthesis ! , EndFraction , EndRoot "P" Subscript, "l" "m" , Baseline left parenthesis, cos phi , right parenthesis sine left parenthesis, "m" psi , right parenthesisYIm=2l+14πl−m!l+m!Plmcosϕsinmψ
equals=
00
12
Expression 13: "W" equals left parenthesis, left parenthesis, StartFraction, 2 Over "n" "a" Subscript, 0 , Baseline , EndFraction , right parenthesis cubed StartFraction, left parenthesis, "n" minus "l" minus 1 , right parenthesis ! Over 2 "n" left parenthesis, left parenthesis, "n" plus "l" , right parenthesis ! , right parenthesis , EndFraction , right parenthesis Superscript, 1 half , Baseline "e" Superscript, negative StartFraction, rho Over 2 , EndFraction , Baseline rho Superscript, "l" , Baseline "L" Subscript, "n" "l" , Baseline left parenthesis, rho , right parenthesis "Y" left brace, "r" greater than or equal to 0 , right braceW=2na03n−l−1!2nn+l!12e−ρ2ρlLnlρYr≥0
13
Expression 14: "W" Subscript, "R" "e" , Baseline equals left parenthesis, left parenthesis, StartFraction, 2 Over "n" "a" Subscript, 0 , Baseline , EndFraction , right parenthesis cubed StartFraction, left parenthesis, "n" minus "l" minus 1 , right parenthesis ! Over 2 "n" left parenthesis, left parenthesis, "n" plus "l" , right parenthesis ! , right parenthesis , EndFraction , right parenthesis Superscript, 1 half , Baseline "e" Superscript, negative StartFraction, rho Over 2 , EndFraction , Baseline rho Superscript, "l" , Baseline "L" Subscript, "n" "l" , Baseline left parenthesis, rho , right parenthesis "Y" Subscript, "R" "e" , Baseline left brace, "r" greater than or equal to 0 , right braceWRe=2na03n−l−1!2nn+l!12e−ρ2ρlLnlρYRer≥0
14
Expression 15: "W" Subscript, "I" "m" , Baseline equals left parenthesis, left parenthesis, StartFraction, 2 Over "n" "a" Subscript, 0 , Baseline , EndFraction , right parenthesis cubed StartFraction, left parenthesis, "n" minus "l" minus 1 , right parenthesis ! Over 2 "n" left parenthesis, left parenthesis, "n" plus "l" , right parenthesis ! , right parenthesis , EndFraction , right parenthesis Superscript, 1 half , Baseline "e" Superscript, negative StartFraction, rho Over 2 , EndFraction , Baseline rho Superscript, "l" , Baseline "L" Subscript, "n" "l" , Baseline left parenthesis, rho , right parenthesis "Y" Subscript, "I" "m" , Baseline left brace, "r" greater than or equal to 0 , right braceWIm=2na03n−l−1!2nn+l!12e−ρ2ρlLnlρYImr≥0
15
Expression 16: "R" Subscript, "D" "F" , Baseline equals 4 pi "r" squared left parenthesis, "W" , right parenthesis squared left brace, "r" greater than or equal to 0 , right braceRDF=4πr2W2r≥0
16
Expression 17: "R" Subscript, "D" "F" , Baseline equals left brace, 0 less than StartAbsoluteValue, "m" psi , EndAbsoluteValue less than pi : 4 pi "r" squared left parenthesis, "W" Subscript, "R" "e" , Baseline , right parenthesis squared , pi less than StartAbsoluteValue, "m" psi , EndAbsoluteValue less than 2 pi : 4 pi "r" squared left parenthesis, "W" Subscript, "R" "e" , Baseline , right parenthesis squared , right brace left brace, "r" greater than or equal to 0 , right braceRDF=0<mψ<π:4πr2WRe2,π<mψ<2π:4πr2WRe2r≥0
17
Expression 18: "R" Subscript, "D" "F" , Baseline equals left brace, 0 less than StartAbsoluteValue, "m" psi , EndAbsoluteValue less than pi : 4 pi "r" squared left parenthesis, "W" Subscript, "I" "m" , Baseline , right parenthesis squared , pi less than StartAbsoluteValue, "m" psi , EndAbsoluteValue less than 2 pi : 4 pi "r" squared left parenthesis, "W" Subscript, "I" "m" , Baseline , right parenthesis squared , right brace left brace, "r" greater than or equal to 0 , right braceRDF=0<mψ<π:4πr2WIm2,π<mψ<2π:4πr2WIm2r≥0