Lamé Functions - 29.6 Fourier Series

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29.6.E5 1 2 ⁢ A 0 2 + ∑ p = 1 ∞ A 2 ⁢ p 2 = 1 1 2 superscript subscript 𝐴 0 2 superscript subscript 𝑝 1 superscript subscript 𝐴 2 𝑝 2 1 {\displaystyle{\displaystyle\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2% }=1}}
\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2} = 1		 

(1)/(2)*(A[0])^(2)+ sum((A[2*p])^(2), p = 1..infinity) = 1
 
Divide[1,2]*(Subscript[A, 0])^(2)+ Sum[(Subscript[A, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1
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29.6.E6 1 2 ⁢ A 0 + ∑ p = 1 ∞ A 2 ⁢ p > 0 1 2 subscript 𝐴 0 superscript subscript 𝑝 1 subscript 𝐴 2 𝑝 0 {\displaystyle{\displaystyle\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p}>0}}
\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p} > 0		 

(1)/(2)*A[0]+ sum(A[2*p], p = 1..infinity) > 0
 
Divide[1,2]*Subscript[A, 0]+ Sum[Subscript[A, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0
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29.6.E7 lim p → ∞ ⁡ A 2 ⁢ p + 2 A 2 ⁢ p = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐴 2 𝑝 2 subscript 𝐴 2 𝑝 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}}=\frac{k^{% 2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n , ν = 2 ⁢ n , m > n formulae-sequence 𝜈 2 𝑛 formulae-sequence 𝜈 2 𝑛 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n,\nu=2n,m>n}} 
limit((A[2*p + 2])/(A[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[A, 2*p + 2],Subscript[A, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6#Ex2 β p = 4 ⁢ p 2 ⁢ ( 2 - k 2 ) subscript 𝛽 𝑝 4 superscript 𝑝 2 2 superscript 𝑘 2 {\displaystyle{\displaystyle\beta_{p}=4p^{2}(2-k^{2})}}
\beta_{p} = 4p^{2}(2-k^{2})		 

beta[p] = 4*(p)^(2)*(2 - (k)^(2))
 
Subscript[\[Beta], p] == 4*(p)^(2)*(2 - (k)^(2))
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29.6#Ex3 γ p = 1 2 ⁢ ( ν - 2 ⁢ p + 1 ) ⁢ ( ν + 2 ⁢ p ) ⁢ k 2 subscript 𝛾 𝑝 1 2 𝜈 2 𝑝 1 𝜈 2 𝑝 superscript 𝑘 2 {\displaystyle{\displaystyle\gamma_{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}}}
\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}		 

((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p + 1)*(nu + 2*p)*(k)^(2)
 
(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p + 1)*(\[Nu]+ 2*p)*(k)^(2)
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29.6.E12 ( 1 - 1 2 ⁢ k 2 ) ⁢ ( 1 2 ⁢ C 0 2 + ∑ p = 1 ∞ C 2 ⁢ p 2 ) - 1 2 ⁢ k 2 ⁢ ∑ p = 0 ∞ C 2 ⁢ p ⁢ C 2 ⁢ p + 2 = 1 1 1 2 superscript 𝑘 2 1 2 superscript subscript 𝐶 0 2 superscript subscript 𝑝 1 superscript subscript 𝐶 2 𝑝 2 1 2 superscript 𝑘 2 superscript subscript 𝑝 0 subscript 𝐶 2 𝑝 subscript 𝐶 2 𝑝 2 1 {\displaystyle{\displaystyle\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}% C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{% \infty}C_{2p}C_{2p+2}=1}}
\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2} = 1		 

(1 -(1)/(2)*(k)^(2))*((1)/(2)*(C[0])^(2)+ sum((C[2*p])^(2), p = 1..infinity))-(1)/(2)*(k)^(2)* sum(C[2*p]*C[2*p + 2], p = 0..infinity) = 1
 
(1 -Divide[1,2]*(k)^(2))*(Divide[1,2]*(Subscript[C, 0])^(2)+ Sum[(Subscript[C, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None])-Divide[1,2]*(k)^(2)* Sum[Subscript[C, 2*p]*Subscript[C, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] == 1
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29.6.E13 1 2 ⁢ C 0 + ∑ p = 1 ∞ C 2 ⁢ p > 0 1 2 subscript 𝐶 0 superscript subscript 𝑝 1 subscript 𝐶 2 𝑝 0 {\displaystyle{\displaystyle\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}>0}}
\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p} > 0		 

(1)/(2)*C[0]+ sum(C[2*p], p = 1..infinity) > 0
 
Divide[1,2]*Subscript[C, 0]+ Sum[Subscript[C, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0
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29.6.E14 lim p → ∞ ⁡ C 2 ⁢ p + 2 C 2 ⁢ p = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐶 2 𝑝 2 subscript 𝐶 2 𝑝 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}}=\frac{k^{% 2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 1 , ν = 2 ⁢ n + 1 , m > n formulae-sequence 𝜈 2 𝑛 1 formulae-sequence 𝜈 2 𝑛 1 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+1,\nu=2n+1,m>n}} 
limit((C[2*p + 2])/(C[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[C, 2*p + 2],Subscript[C, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6.E20 ∑ p = 0 ∞ A 2 ⁢ p + 1 2 = 1 superscript subscript 𝑝 0 superscript subscript 𝐴 2 𝑝 1 2 1 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}A_{2p+1}^{2}=1}}
\sum_{p=0}^{\infty}A_{2p+1}^{2} = 1		 

sum((A[2*p + 1])^(2), p = 0..infinity) = 1
 
Sum[(Subscript[A, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1
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29.6.E21 ∑ p = 0 ∞ A 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 subscript 𝐴 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}A_{2p+1}>0}}
\sum_{p=0}^{\infty}A_{2p+1} > 0		 

sum(A[2*p + 1], p = 0..infinity) > 0
 
Sum[Subscript[A, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E22 lim p → ∞ ⁡ A 2 ⁢ p + 1 A 2 ⁢ p - 1 = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐴 2 𝑝 1 subscript 𝐴 2 𝑝 1 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}}=\frac{k% ^{2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 1 , ν = 2 ⁢ n + 1 , m > n formulae-sequence 𝜈 2 𝑛 1 formulae-sequence 𝜈 2 𝑛 1 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+1,\nu=2n+1,m>n}} 
limit((A[2*p + 1])/(A[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[A, 2*p + 1],Subscript[A, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6#Ex4 α p = 1 2 ⁢ ( ν - 2 ⁢ p - 1 ) ⁢ ( ν + 2 ⁢ p + 2 ) ⁢ k 2 subscript 𝛼 𝑝 1 2 𝜈 2 𝑝 1 𝜈 2 𝑝 2 superscript 𝑘 2 {\displaystyle{\displaystyle\alpha_{p}=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}}}
\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}		 

alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)
 
Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)
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29.6#Ex6 γ p = 1 2 ⁢ ( ν - 2 ⁢ p ) ⁢ ( ν + 2 ⁢ p + 1 ) ⁢ k 2 subscript 𝛾 𝑝 1 2 𝜈 2 𝑝 𝜈 2 𝑝 1 superscript 𝑘 2 {\displaystyle{\displaystyle\gamma_{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}}}
\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}		 

((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)
 
(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)
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29.6.E28 ∑ p = 0 ∞ C 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 subscript 𝐶 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}C_{2p+1}>0}}
\sum_{p=0}^{\infty}C_{2p+1} > 0		 

sum(C[2*p + 1], p = 0..infinity) > 0
 
Sum[Subscript[C, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E29 lim p → ∞ ⁡ C 2 ⁢ p + 1 C 2 ⁢ p - 1 = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐶 2 𝑝 1 subscript 𝐶 2 𝑝 1 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}}=\frac{k% ^{2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 2 , ν = 2 ⁢ n + 2 , m > n formulae-sequence 𝜈 2 𝑛 2 formulae-sequence 𝜈 2 𝑛 2 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+2,\nu=2n+2,m>n}} 
limit((C[2*p + 1])/(C[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[C, 2*p + 1],Subscript[C, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6.E35 ∑ p = 0 ∞ B 2 ⁢ p + 1 2 = 1 superscript subscript 𝑝 0 superscript subscript 𝐵 2 𝑝 1 2 1 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}B_{2p+1}^{2}=1}}
\sum_{p=0}^{\infty}B_{2p+1}^{2} = 1		 

sum((B[2*p + 1])^(2), p = 0..infinity) = 1
 
Sum[(Subscript[B, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1
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29.6.E36 ∑ p = 0 ∞ ( 2 ⁢ p + 1 ) ⁢ B 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 2 𝑝 1 subscript 𝐵 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}(2p+1)B_{2p+1}>0}}
\sum_{p=0}^{\infty}(2p+1)B_{2p+1} > 0		 

sum((2*p + 1)*B[2*p + 1], p = 0..infinity) > 0
 
Sum[(2*p + 1)*Subscript[B, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E37 lim p → ∞ ⁡ B 2 ⁢ p + 1 B 2 ⁢ p - 1 = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐵 2 𝑝 1 subscript 𝐵 2 𝑝 1 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}}=\frac{k% ^{2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 1 , ν = 2 ⁢ n + 1 , m > n formulae-sequence 𝜈 2 𝑛 1 formulae-sequence 𝜈 2 𝑛 1 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+1,\nu=2n+1,m>n}} 
limit((B[2*p + 1])/(B[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[B, 2*p + 1],Subscript[B, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6#Ex7 α p = 1 2 ⁢ ( ν - 2 ⁢ p - 1 ) ⁢ ( ν + 2 ⁢ p + 2 ) ⁢ k 2 subscript 𝛼 𝑝 1 2 𝜈 2 𝑝 1 𝜈 2 𝑝 2 superscript 𝑘 2 {\displaystyle{\displaystyle\alpha_{p}=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}}}
\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}		 

alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)
 
Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)
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29.6#Ex9 γ p = 1 2 ⁢ ( ν - 2 ⁢ p ) ⁢ ( ν + 2 ⁢ p + 1 ) ⁢ k 2 subscript 𝛾 𝑝 1 2 𝜈 2 𝑝 𝜈 2 𝑝 1 superscript 𝑘 2 {\displaystyle{\displaystyle\gamma_{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}}}
\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}		 

((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)
 
(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)
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29.6.E43 ∑ p = 0 ∞ ( 2 ⁢ p + 1 ) ⁢ D 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 2 𝑝 1 subscript 𝐷 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0}}
\sum_{p=0}^{\infty}(2p+1)D_{2p+1} > 0		 

sum((2*p + 1)*D[2*p + 1], p = 0..infinity) > 0
 
Sum[(2*p + 1)*Subscript[D, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E44 lim p → ∞ ⁡ D 2 ⁢ p + 1 D 2 ⁢ p - 1 = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐷 2 𝑝 1 subscript 𝐷 2 𝑝 1 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}}=\frac{k% ^{2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 2 , ν = 2 ⁢ n + 2 , m > n formulae-sequence 𝜈 2 𝑛 2 formulae-sequence 𝜈 2 𝑛 2 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+2,\nu=2n+2,m>n}} 
limit((D[2*p + 1])/(D[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[D, 2*p + 1],Subscript[D, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6.E50 ∑ p = 1 ∞ B 2 ⁢ p 2 = 1 superscript subscript 𝑝 1 superscript subscript 𝐵 2 𝑝 2 1 {\displaystyle{\displaystyle\sum_{p=1}^{\infty}B_{2p}^{2}=1}}
\sum_{p=1}^{\infty}B_{2p}^{2} = 1		 

sum((B[2*p])^(2), p = 1..infinity) = 1
 
Sum[(Subscript[B, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1
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29.6.E51 ∑ p = 0 ∞ ( 2 ⁢ p + 2 ) ⁢ B 2 ⁢ p + 2 > 0 superscript subscript 𝑝 0 2 𝑝 2 subscript 𝐵 2 𝑝 2 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}(2p+2)B_{2p+2}>0}}
\sum_{p=0}^{\infty}(2p+2)B_{2p+2} > 0		 

sum((2*p + 2)*B[2*p + 2], p = 0..infinity) > 0
 
Sum[(2*p + 2)*Subscript[B, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E52 lim p → ∞ ⁡ B 2 ⁢ p + 2 B 2 ⁢ p = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐵 2 𝑝 2 subscript 𝐵 2 𝑝 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}}=\frac{k^{% 2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 2 , ν = 2 ⁢ n + 2 , m > n formulae-sequence 𝜈 2 𝑛 2 formulae-sequence 𝜈 2 𝑛 2 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+2,\nu=2n+2,m>n}} 
limit((B[2*p + 2])/(B[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[B, 2*p + 2],Subscript[B, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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29.6#Ex10 α p = 1 2 ⁢ ( ν - 2 ⁢ p - 2 ) ⁢ ( ν + 2 ⁢ p + 3 ) ⁢ k 2 subscript 𝛼 𝑝 1 2 𝜈 2 𝑝 2 𝜈 2 𝑝 3 superscript 𝑘 2 {\displaystyle{\displaystyle\alpha_{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}}}
\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}		 

alpha[p] = (1)/(2)*(nu - 2*p - 2)*(nu + 2*p + 3)*(k)^(2)
 
Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 2)*(\[Nu]+ 2*p + 3)*(k)^(2)
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29.6#Ex11 β p = ( 2 ⁢ p + 2 ) 2 ⁢ ( 2 - k 2 ) subscript 𝛽 𝑝 superscript 2 𝑝 2 2 2 superscript 𝑘 2 {\displaystyle{\displaystyle\beta_{p}=(2p+2)^{2}(2-k^{2})}}
\beta_{p} = (2p+2)^{2}(2-k^{2})		 

beta[p] = (2*p + 2)^(2)*(2 - (k)^(2))
 
Subscript[\[Beta], p] == (2*p + 2)^(2)*(2 - (k)^(2))
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29.6#Ex12 γ p = 1 2 ⁢ ( ν - 2 ⁢ p - 1 ) ⁢ ( ν + 2 ⁢ p + 2 ) ⁢ k 2 subscript 𝛾 𝑝 1 2 𝜈 2 𝑝 1 𝜈 2 𝑝 2 superscript 𝑘 2 {\displaystyle{\displaystyle\gamma_{p}=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}}}
\gamma_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}		 

((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)
 
(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)
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29.6.E57 ( 1 - 1 2 ⁢ k 2 ) ⁢ ∑ p = 1 ∞ D 2 ⁢ p 2 - 1 2 ⁢ k 2 ⁢ ∑ p = 1 ∞ D 2 ⁢ p ⁢ D 2 ⁢ p + 2 = 1 1 1 2 superscript 𝑘 2 superscript subscript 𝑝 1 superscript subscript 𝐷 2 𝑝 2 1 2 superscript 𝑘 2 superscript subscript 𝑝 1 subscript 𝐷 2 𝑝 subscript 𝐷 2 𝑝 2 1 {\displaystyle{\displaystyle\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty% }D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2}=1}}
\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2} = 1		 

(1 -(1)/(2)*(k)^(2))*sum((D[2*p])^(2), p = 1..infinity)-(1)/(2)*(k)^(2)* sum(D[2*p]*D[2*p + 2], p = 1..infinity) = 1
 
(1 -Divide[1,2]*(k)^(2))*Sum[(Subscript[D, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None]-Divide[1,2]*(k)^(2)* Sum[Subscript[D, 2*p]*Subscript[D, 2*p + 2], {p, 1, Infinity}, GenerateConditions->None] == 1
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29.6.E58 ∑ p = 0 ∞ ( 2 ⁢ p + 2 ) ⁢ D 2 ⁢ p + 2 > 0 superscript subscript 𝑝 0 2 𝑝 2 subscript 𝐷 2 𝑝 2 0 {\displaystyle{\displaystyle\sum_{p=0}^{\infty}(2p+2)D_{2p+2}>0}}
\sum_{p=0}^{\infty}(2p+2)D_{2p+2} > 0		 

sum((2*p + 2)*D[2*p + 2], p = 0..infinity) > 0
 
Sum[(2*p + 2)*Subscript[D, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0
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29.6.E59 lim p → ∞ ⁡ D 2 ⁢ p + 2 D 2 ⁢ p = k 2 ( 1 + k ′ ) 2 subscript → 𝑝 subscript 𝐷 2 𝑝 2 subscript 𝐷 2 𝑝 superscript 𝑘 2 superscript 1 superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}}=\frac{k^{% 2}}{(1+k^{\prime})^{2}}}}
\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}		 ν ≠ 2 ⁢ n + 3 , ν = 2 ⁢ n + 3 , m > n formulae-sequence 𝜈 2 𝑛 3 formulae-sequence 𝜈 2 𝑛 3 𝑚 𝑛 {\displaystyle{\displaystyle\nu\neq 2n+3,\nu=2n+3,m>n}} 
limit((D[2*p + 2])/(D[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))
 
Limit[Divide[Subscript[D, 2*p + 2],Subscript[D, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]
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