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 A j , j = ( m + 2 ⁢ j - 2 ) ⁢ ( m + 2 ⁢ j - 1 ) - 2 ⁢ γ 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 superscript 𝛾 2 𝑚 2 𝑗 2 𝑚 2 𝑗 1 1 superscript 𝑚 2 2 𝑚 4 𝑗 5 2 𝑚 4 𝑗 1 {\displaystyle{\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)}		 

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)]
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30.16#Ex2 A j , j + 1 = - γ 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 superscript 𝛾 2 2 𝑚 2 𝑗 1 2 𝑚 2 𝑗 2 𝑚 4 𝑗 1 2 𝑚 4 𝑗 1 {\displaystyle{\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)}		 

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)]
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30.16#Ex3 A j , j - 1 = - γ 2 ⁢ ( 2 ⁢ j - 3 ) ⁢ ( 2 ⁢ j - 2 ) ( 2 ⁢ m + 4 ⁢ j - 7 ) ⁢ ( 2 ⁢ m + 4 ⁢ j - 5 ) subscript 𝐴 𝑗 𝑗 1 superscript 𝛾 2 2 𝑗 3 2 𝑗 2 2 𝑚 4 𝑗 7 2 𝑚 4 𝑗 5 {\displaystyle{\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)}		 

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)]
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30.16.E2 α j , d + 1 ≤ α j , d subscript 𝛼 𝑗 𝑑 1 subscript 𝛼 𝑗 𝑑 {\displaystyle{\displaystyle\alpha_{j,d+1}\leq\alpha_{j,d}}}
\alpha_{j,d+1} \leq \alpha_{j,d}		 

alpha[j , d + 1] <= alpha[j , d]
 
Subscript[\[Alpha], j , d + 1] <= Subscript[\[Alpha], j , d]
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30.16#Ex4 α 2 , 2 = 14.18833 246 subscript 𝛼 2 2 14.18833 246 {\displaystyle{\displaystyle\alpha_{2,2}=14.18833\;246}}
\alpha_{2,2} = 14.18833\;246		 

alpha[2 , 2] = 14.18833246
 
Subscript[\[Alpha], 2 , 2] == 14.18833246
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30.16#Ex5 α 2 , 3 = 13.98002 013 subscript 𝛼 2 3 13.98002 013 {\displaystyle{\displaystyle\alpha_{2,3}=13.98002\;013}}
\alpha_{2,3} = 13.98002\;013		 

alpha[2 , 3] = 13.98002013
 
Subscript[\[Alpha], 2 , 3] == 13.98002013
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30.16#Ex6 α 2 , 4 = 13.97907 459 subscript 𝛼 2 4 13.97907 459 {\displaystyle{\displaystyle\alpha_{2,4}=13.97907\;459}}
\alpha_{2,4} = 13.97907\;459		 

alpha[2 , 4] = 13.97907459
 
Subscript[\[Alpha], 2 , 4] == 13.97907459
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30.16#Ex7 α 2 , 5 = 13.97907 345 subscript 𝛼 2 5 13.97907 345 {\displaystyle{\displaystyle\alpha_{2,5}=13.97907\;345}}
\alpha_{2,5} = 13.97907\;345		 

alpha[2 , 5] = 13.97907345
 
Subscript[\[Alpha], 2 , 5] == 13.97907345
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30.16#Ex8 α 2 , 6 = 13.97907 345 subscript 𝛼 2 6 13.97907 345 {\displaystyle{\displaystyle\alpha_{2,6}=13.97907\;345}}
\alpha_{2,6} = 13.97907\;345		 

alpha[2 , 6] = 13.97907345
 
Subscript[\[Alpha], 2 , 6] == 13.97907345
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30.16#Ex9 A j , j = ( m + 2 ⁢ j - 1 ) ⁢ ( m + 2 ⁢ j ) - 2 ⁢ γ 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 superscript 𝛾 2 𝑚 2 𝑗 1 𝑚 2 𝑗 1 superscript 𝑚 2 2 𝑚 4 𝑗 3 2 𝑚 4 𝑗 1 {\displaystyle{\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)}		 

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)]
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30.16#Ex10 A j , j + 1 = - γ 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 superscript 𝛾 2 2 𝑚 2 𝑗 2 𝑚 2 𝑗 1 2 𝑚 4 𝑗 1 2 𝑚 4 𝑗 3 {\displaystyle{\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)}		 

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)]
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30.16#Ex11 A j , j - 1 = - γ 2 ⁢ ( 2 ⁢ j - 2 ) ⁢ ( 2 ⁢ j - 1 ) ( 2 ⁢ m + 4 ⁢ j - 5 ) ⁢ ( 2 ⁢ m + 4 ⁢ j - 3 ) subscript 𝐴 𝑗 𝑗 1 superscript 𝛾 2 2 𝑗 2 2 𝑗 1 2 𝑚 4 𝑗 5 2 𝑚 4 𝑗 3 {\displaystyle{\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)}		 

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)]
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30.16.E7 ∑ j = 1 d e j , d 2 ⁢ ( n + m + 2 ⁢ j - 2 ⁢ p ) ! ( n - m + 2 ⁢ j - 2 ⁢ p ) ! ⁢ 1 2 ⁢ n + 4 ⁢ j - 4 ⁢ p + 1 = ( n + m ) ! ( n - m ) ! ⁢ 1 2 ⁢ n + 1 superscript subscript 𝑗 1 𝑑 superscript subscript 𝑒 𝑗 𝑑 2 𝑛 𝑚 2 𝑗 2 𝑝 𝑛 𝑚 2 𝑗 2 𝑝 1 2 𝑛 4 𝑗 4 𝑝 1 𝑛 𝑚 𝑛 𝑚 1 2 𝑛 1 {\displaystyle{\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}		 

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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