Spheroidal Wave Functions - 30.16 Methods of Computation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
30.16#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j} = (m+2j-2)(m+2j-1)-2\gamma^{2}\frac{(m+2j-2)(m+2j-1)-1+m^{2}}{(2m+4j-5)(2m+4j-1)}}
A_{j,j} = (m+2j-2)(m+2j-1)-2\gamma^{2}\frac{(m+2j-2)(m+2j-1)-1+m^{2}}{(2m+4j-5)(2m+4j-1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j] = (m + 2*j - 2)*(m + 2*j - 1)- 2*(gamma)^(2)*((m + 2*j - 2)*(m + 2*j - 1)- 1 + (m)^(2))/((2*m + 4*j - 5)*(2*m + 4*j - 1))
 
Subscript[A, j , j] == (m + 2*j - 2)*(m + 2*j - 1)- 2*\[Gamma]^(2)*Divide[(m + 2*j - 2)*(m + 2*j - 1)- 1 + (m)^(2),(2*m + 4*j - 5)*(2*m + 4*j - 1)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j+1} = -\gamma^{2}\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)}}
A_{j,j+1} = -\gamma^{2}\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j + 1] = - (gamma)^(2)*((2*m + 2*j - 1)*(2*m + 2*j))/((2*m + 4*j - 1)*(2*m + 4*j + 1))
 
Subscript[A, j , j + 1] == - \[Gamma]^(2)*Divide[(2*m + 2*j - 1)*(2*m + 2*j),(2*m + 4*j - 1)*(2*m + 4*j + 1)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j-1} = -\gamma^{2}\frac{(2j-3)(2j-2)}{(2m+4j-7)(2m+4j-5)}}
A_{j,j-1} = -\gamma^{2}\frac{(2j-3)(2j-2)}{(2m+4j-7)(2m+4j-5)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j - 1] = - (gamma)^(2)*((2*j - 3)*(2*j - 2))/((2*m + 4*j - 7)*(2*m + 4*j - 5))
 
Subscript[A, j , j - 1] == - \[Gamma]^(2)*Divide[(2*j - 3)*(2*j - 2),(2*m + 4*j - 7)*(2*m + 4*j - 5)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{j,d+1} \leq \alpha_{j,d}}
\alpha_{j,d+1} \leq \alpha_{j,d}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[j , d + 1] <= alpha[j , d]
 
Subscript[\[Alpha], j , d + 1] <= Subscript[\[Alpha], j , d]
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{2,2} = 14.18833\;246}
\alpha_{2,2} = 14.18833\;246		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[2 , 2] = 14.18833246
 
Subscript[\[Alpha], 2 , 2] == 14.18833246
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{2,3} = 13.98002\;013}
\alpha_{2,3} = 13.98002\;013		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[2 , 3] = 13.98002013
 
Subscript[\[Alpha], 2 , 3] == 13.98002013
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{2,4} = 13.97907\;459}
\alpha_{2,4} = 13.97907\;459		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[2 , 4] = 13.97907459
 
Subscript[\[Alpha], 2 , 4] == 13.97907459
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{2,5} = 13.97907\;345}
\alpha_{2,5} = 13.97907\;345		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[2 , 5] = 13.97907345
 
Subscript[\[Alpha], 2 , 5] == 13.97907345
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{2,6} = 13.97907\;345}
\alpha_{2,6} = 13.97907\;345		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[2 , 6] = 13.97907345
 
Subscript[\[Alpha], 2 , 6] == 13.97907345
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j} = (m+2j-1)(m+2j)-2\gamma^{2}\*\frac{(m+2j-1)(m+2j)-1+m^{2}}{(2m+4j-3)(2m+4j+1)}}
A_{j,j} = (m+2j-1)(m+2j)-2\gamma^{2}\*\frac{(m+2j-1)(m+2j)-1+m^{2}}{(2m+4j-3)(2m+4j+1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j] = (m + 2*j - 1)*(m + 2*j)- 2*(gamma)^(2)*((m + 2*j - 1)*(m + 2*j)- 1 + (m)^(2))/((2*m + 4*j - 3)*(2*m + 4*j + 1))
 
Subscript[A, j , j] == (m + 2*j - 1)*(m + 2*j)- 2*\[Gamma]^(2)*Divide[(m + 2*j - 1)*(m + 2*j)- 1 + (m)^(2),(2*m + 4*j - 3)*(2*m + 4*j + 1)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j+1} = -\gamma^{2}\frac{(2m+2j)(2m+2j+1)}{(2m+4j+1)(2m+4j+3)}}
A_{j,j+1} = -\gamma^{2}\frac{(2m+2j)(2m+2j+1)}{(2m+4j+1)(2m+4j+3)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j + 1] = - (gamma)^(2)*((2*m + 2*j)*(2*m + 2*j + 1))/((2*m + 4*j + 1)*(2*m + 4*j + 3))
 
Subscript[A, j , j + 1] == - \[Gamma]^(2)*Divide[(2*m + 2*j)*(2*m + 2*j + 1),(2*m + 4*j + 1)*(2*m + 4*j + 3)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{j,j-1} = -\gamma^{2}\frac{(2j-2)(2j-1)}{(2m+4j-5)(2m+4j-3)}}
A_{j,j-1} = -\gamma^{2}\frac{(2j-2)(2j-1)}{(2m+4j-5)(2m+4j-3)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A[j , j - 1] = - (gamma)^(2)*((2*j - 2)*(2*j - 1))/((2*m + 4*j - 5)*(2*m + 4*j - 3))
 
Subscript[A, j , j - 1] == - \[Gamma]^(2)*Divide[(2*j - 2)*(2*j - 1),(2*m + 4*j - 5)*(2*m + 4*j - 3)]
 Skipped - no semantic math Skipped - no semantic math - -
30.16.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=1}^{d}e_{j,d}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1} = \frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}}
\sum_{j=1}^{d}e_{j,d}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1} = \frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum((exp(1)[j , d])^(2)*(factorial(n + m + 2*j - 2*p))/(factorial(n - m + 2*j - 2*p))*(1)/(2*n + 4*j - 4*p + 1), j = 1..d) = (factorial(n + m))/(factorial(n - m))*(1)/(2*n + 1)
 
Sum[(Subscript[E, j , d])^(2)*Divide[(n + m + 2*j - 2*p)!,(n - m + 2*j - 2*p)!]*Divide[1,2*n + 4*j - 4*p + 1], {j, 1, d}, GenerateConditions->None] == Divide[(n + m)!,(n - m)!]*Divide[1,2*n + 1]
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