Spheroidal Wave Functions - 30.3 Eigenvalues

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30.3#Ex4 α k = γ 2 ( k + 2 m + 1 ) ( k + 2 m + 2 ) ( 2 k + 2 m + 3 ) ( 2 k + 2 m + 5 ) subscript 𝛼 𝑘 superscript 𝛾 2 𝑘 2 𝑚 1 𝑘 2 𝑚 2 2 𝑘 2 𝑚 3 2 𝑘 2 𝑚 5 {\displaystyle{\displaystyle\alpha_{k}=\gamma^{2}\frac{(k+2m+1)(k+2m+2)}{(2k+2% m+3)(2k+2m+5)}}}
\alpha_{k} = \gamma^{2}\frac{(k+2m+1)(k+2m+2)}{(2k+2m+3)(2k+2m+5)}

alpha[k] = (gamma)^(2)*((k + 2*m + 1)*(k + 2*m + 2))/((2*k + 2*m + 3)*(2*k + 2*m + 5))
Subscript[\[Alpha], k] == \[Gamma]^(2)*Divide[(k + 2*m + 1)*(k + 2*m + 2),(2*k + 2*m + 3)*(2*k + 2*m + 5)]
Skipped - no semantic math Skipped - no semantic math - -
30.3#Ex5 β k = ( k + m ) ( k + m + 1 ) - 2 γ 2 ( k + m ) ( k + m + 1 ) - 1 + m 2 ( 2 k + 2 m - 1 ) ( 2 k + 2 m + 3 ) subscript 𝛽 𝑘 𝑘 𝑚 𝑘 𝑚 1 2 superscript 𝛾 2 𝑘 𝑚 𝑘 𝑚 1 1 superscript 𝑚 2 2 𝑘 2 𝑚 1 2 𝑘 2 𝑚 3 {\displaystyle{\displaystyle\beta_{k}=(k+m)(k+m+1)-2\gamma^{2}\frac{(k+m)(k+m+% 1)-1+m^{2}}{(2k+2m-1)(2k+2m+3)}}}
\beta_{k} = (k+m)(k+m+1)-2\gamma^{2}\frac{(k+m)(k+m+1)-1+m^{2}}{(2k+2m-1)(2k+2m+3)}

beta[k] = (k + m)*(k + m + 1)- 2*(gamma)^(2)*((k + m)*(k + m + 1)- 1 + (m)^(2))/((2*k + 2*m - 1)*(2*k + 2*m + 3))
Subscript[\[Beta], k] == (k + m)*(k + m + 1)- 2*\[Gamma]^(2)*Divide[(k + m)*(k + m + 1)- 1 + (m)^(2),(2*k + 2*m - 1)*(2*k + 2*m + 3)]
Skipped - no semantic math Skipped - no semantic math - -
30.3#Ex6 γ k = γ 2 ( k - 1 ) k ( 2 k + 2 m - 3 ) ( 2 k + 2 m - 1 ) subscript 𝛾 𝑘 superscript 𝛾 2 𝑘 1 𝑘 2 𝑘 2 𝑚 3 2 𝑘 2 𝑚 1 {\displaystyle{\displaystyle\gamma_{k}=\gamma^{2}\frac{(k-1)k}{(2k+2m-3)(2k+2m% -1)}}}
\gamma_{k} = \gamma^{2}\frac{(k-1)k}{(2k+2m-3)(2k+2m-1)}

((gamma)^(2)) = (gamma)^(2)*((k - 1)*k)/((2*k + 2*m - 3)*(2*k + 2*m - 1))
(\[Gamma]^(2)) == \[Gamma]^(2)*Divide[(k - 1)*k,(2*k + 2*m - 3)*(2*k + 2*m - 1)]
Skipped - no semantic math Skipped - no semantic math - -
30.3#Ex7 0 = n ( n + 1 ) subscript 0 𝑛 𝑛 1 {\displaystyle{\displaystyle\ell_{0}=n(n+1)}}
\ell_{0} = n(n+1)

ell[0] = n*(n + 1)
Subscript[\[ScriptL], 0] == n*(n + 1)
Skipped - no semantic math Skipped - no semantic math - -
30.3#Ex8 2 2 = - 1 - ( 2 m - 1 ) ( 2 m + 1 ) ( 2 n - 1 ) ( 2 n + 3 ) 2 subscript 2 1 2 𝑚 1 2 𝑚 1 2 𝑛 1 2 𝑛 3 {\displaystyle{\displaystyle 2\ell_{2}=-1-\frac{(2m-1)(2m+1)}{(2n-1)(2n+3)}}}
2\ell_{2} = -1-\frac{(2m-1)(2m+1)}{(2n-1)(2n+3)}

2*ell[2] = - 1 -((2*m - 1)*(2*m + 1))/((2*n - 1)*(2*n + 3))
2*Subscript[\[ScriptL], 2] == - 1 -Divide[(2*m - 1)*(2*m + 1),(2*n - 1)*(2*n + 3)]
Skipped - no semantic math Skipped - no semantic math - -
30.3#Ex9 2 4 = ( n - m - 1 ) ( n - m ) ( n + m - 1 ) ( n + m ) ( 2 n - 3 ) ( 2 n - 1 ) 3 ( 2 n + 1 ) - ( n - m + 1 ) ( n - m + 2 ) ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) 3 ( 2 n + 5 ) 2 subscript 4 𝑛 𝑚 1 𝑛 𝑚 𝑛 𝑚 1 𝑛 𝑚 2 𝑛 3 superscript 2 𝑛 1 3 2 𝑛 1 𝑛 𝑚 1 𝑛 𝑚 2 𝑛 𝑚 1 𝑛 𝑚 2 2 𝑛 1 superscript 2 𝑛 3 3 2 𝑛 5 {\displaystyle{\displaystyle 2\ell_{4}=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-3)(% 2n-1)^{3}(2n+1)}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n+1)(2n+3)^{3}(2n+5)}}}
2\ell_{4} = \frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-3)(2n-1)^{3}(2n+1)}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n+1)(2n+3)^{3}(2n+5)}

2*ell[4] = ((n - m - 1)*(n - m)*(n + m - 1)*(n + m))/((2*n - 3)*(2*n - 1)^(3)*(2*n + 1))-((n - m + 1)*(n - m + 2)*(n + m + 1)*(n + m + 2))/((2*n + 1)*(2*n + 3)^(3)*(2*n + 5))
2*Subscript[\[ScriptL], 4] == Divide[(n - m - 1)*(n - m)*(n + m - 1)*(n + m),(2*n - 3)*(2*n - 1)^(3)*(2*n + 1)]-Divide[(n - m + 1)*(n - m + 2)*(n + m + 1)*(n + m + 2),(2*n + 1)*(2*n + 3)^(3)*(2*n + 5)]
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30.3.E10 6 = ( 4 m 2 - 1 ) ( ( n - m + 1 ) ( n - m + 2 ) ( n + m + 1 ) ( n + m + 2 ) ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) 5 ( 2 n + 5 ) ( 2 n + 7 ) - ( n - m - 1 ) ( n - m ) ( n + m - 1 ) ( n + m ) ( 2 n - 5 ) ( 2 n - 3 ) ( 2 n - 1 ) 5 ( 2 n + 1 ) ( 2 n + 3 ) ) subscript 6 4 superscript 𝑚 2 1 𝑛 𝑚 1 𝑛 𝑚 2 𝑛 𝑚 1 𝑛 𝑚 2 2 𝑛 1 2 𝑛 1 superscript 2 𝑛 3 5 2 𝑛 5 2 𝑛 7 𝑛 𝑚 1 𝑛 𝑚 𝑛 𝑚 1 𝑛 𝑚 2 𝑛 5 2 𝑛 3 superscript 2 𝑛 1 5 2 𝑛 1 2 𝑛 3 {\displaystyle{\displaystyle\ell_{6}=(4m^{2}-1)\left(\frac{(n-m+1)(n-m+2)(n+m+% 1)(n+m+2)}{(2n-1)(2n+1)(2n+3)^{5}(2n+5)(2n+7)}-\frac{(n-m-1)(n-m)(n+m-1)(n+m)}% {(2n-5)(2n-3)(2n-1)^{5}(2n+1)(2n+3)}\right)}}
\ell_{6} = (4m^{2}-1)\left(\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)(2n+1)(2n+3)^{5}(2n+5)(2n+7)}-\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)(2n-3)(2n-1)^{5}(2n+1)(2n+3)}\right)

ell[6] = (4*(m)^(2)- 1)*(((n - m + 1)*(n - m + 2)*(n + m + 1)*(n + m + 2))/((2*n - 1)*(2*n + 1)*(2*n + 3)^(5)*(2*n + 5)*(2*n + 7))-((n - m - 1)*(n - m)*(n + m - 1)*(n + m))/((2*n - 5)*(2*n - 3)*(2*n - 1)^(5)*(2*n + 1)*(2*n + 3)))
Subscript[\[ScriptL], 6] == (4*(m)^(2)- 1)*(Divide[(n - m + 1)*(n - m + 2)*(n + m + 1)*(n + m + 2),(2*n - 1)*(2*n + 1)*(2*n + 3)^(5)*(2*n + 5)*(2*n + 7)]-Divide[(n - m - 1)*(n - m)*(n + m - 1)*(n + m),(2*n - 5)*(2*n - 3)*(2*n - 1)^(5)*(2*n + 1)*(2*n + 3)])
Skipped - no semantic math Skipped - no semantic math - -
30.3.E11 8 = 2 ( 4 m 2 - 1 ) 2 A + 1 16 B + 1 8 C + 1 2 D subscript 8 2 superscript 4 superscript 𝑚 2 1 2 𝐴 1 16 𝐵 1 8 𝐶 1 2 𝐷 {\displaystyle{\displaystyle\ell_{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8% }C+\frac{1}{2}D}}
\ell_{8} = 2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D

ell[8] = 2*(4*(m)^(2)- 1)^(2)* A +(1)/(16)*B +(1)/(8)*C +(1)/(2)*(((n - m - 1)*(n - m)*(n - m + 1)*(n - m + 2)*(n + m - 1)*(n + m)*(n + m + 1)*(n + m + 2))/((2*n - 3)*(2*n - 1)^(4)*(2*n + 1)^(2)*(2*n + 3)^(4)*(2*n + 5)))
Subscript[\[ScriptL], 8] == 2*(4*(m)^(2)- 1)^(2)* A +Divide[1,16]*B +Divide[1,8]*C +Divide[1,2]*(Divide[(n - m - 1)*(n - m)*(n - m + 1)*(n - m + 2)*(n + m - 1)*(n + m)*(n + m + 1)*(n + m + 2),(2*n - 3)*(2*n - 1)^(4)*(2*n + 1)^(2)*(2*n + 3)^(4)*(2*n + 5)])
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