Lamé Functions - 29.15 Fourier Series and Chebyshev Series

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
29.15.E5 1 2 ⁢ A 0 2 + ∑ p = 1 n A 2 ⁢ p 2 = 1 1 2 superscript subscript 𝐴 0 2 superscript subscript 𝑝 1 𝑛 superscript subscript 𝐴 2 𝑝 2 1 {\displaystyle{\displaystyle\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2}=1}}
\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2} = 1		 

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

(1)/(2)*A[0]+ sum(A[2*p], p = 1..n) > 0
 
Divide[1,2]*Subscript[A, 0]+ Sum[Subscript[A, 2*p], {p, 1, n}, GenerateConditions->None] > 0
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29.15.E10 ∑ p = 0 n A 2 ⁢ p + 1 2 = 1 superscript subscript 𝑝 0 𝑛 superscript subscript 𝐴 2 𝑝 1 2 1 {\displaystyle{\displaystyle\sum_{p=0}^{n}A_{2p+1}^{2}=1}}
\sum_{p=0}^{n}A_{2p+1}^{2} = 1		 

sum((A[2*p + 1])^(2), p = 0..n) = 1
 
Sum[(Subscript[A, 2*p + 1])^(2), {p, 0, n}, GenerateConditions->None] == 1
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29.15.E11 ∑ p = 0 n A 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 𝑛 subscript 𝐴 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}A_{2p+1}>0}}
\sum_{p=0}^{n}A_{2p+1} > 0		 

sum(A[2*p + 1], p = 0..n) > 0
 
Sum[Subscript[A, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0
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29.15.E15 ∑ p = 0 n B 2 ⁢ p + 1 2 = 1 superscript subscript 𝑝 0 𝑛 superscript subscript 𝐵 2 𝑝 1 2 1 {\displaystyle{\displaystyle\sum_{p=0}^{n}B_{2p+1}^{2}=1}}
\sum_{p=0}^{n}B_{2p+1}^{2} = 1		 

sum((B[2*p + 1])^(2), p = 0..n) = 1
 
Sum[(Subscript[B, 2*p + 1])^(2), {p, 0, n}, GenerateConditions->None] == 1
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29.15.E16 ∑ p = 0 n ( 2 ⁢ p + 1 ) ⁢ B 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 𝑛 2 𝑝 1 subscript 𝐵 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}(2p+1)B_{2p+1}>0}}
\sum_{p=0}^{n}(2p+1)B_{2p+1} > 0		 

sum((2*p + 1)*B[2*p + 1], p = 0..n) > 0
 
Sum[(2*p + 1)*Subscript[B, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0
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29.15.E20 ( 1 - 1 2 ⁢ k 2 ) ⁢ ( 1 2 ⁢ C 0 2 + ∑ p = 1 n C 2 ⁢ p 2 ) - 1 2 ⁢ k 2 ⁢ ∑ p = 0 n - 1 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 𝑛 superscript subscript 𝐶 2 𝑝 2 1 2 superscript 𝑘 2 superscript subscript 𝑝 0 𝑛 1 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}^{n}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{% 2p}C_{2p+2}=1}}
\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{n}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{2p}C_{2p+2} = 1		 

(1 -(1)/(2)*(k)^(2))*((1)/(2)*(C[0])^(2)+ sum((C[2*p])^(2), p = 1..n))-(1)/(2)*(k)^(2)* sum(C[2*p]*C[2*p + 2], p = 0..n - 1) = 1
 
(1 -Divide[1,2]*(k)^(2))*(Divide[1,2]*(Subscript[C, 0])^(2)+ Sum[(Subscript[C, 2*p])^(2), {p, 1, n}, GenerateConditions->None])-Divide[1,2]*(k)^(2)* Sum[Subscript[C, 2*p]*Subscript[C, 2*p + 2], {p, 0, n - 1}, GenerateConditions->None] == 1
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29.15.E21 1 2 ⁢ C 0 + ∑ p = 1 n C 2 ⁢ p > 0 1 2 subscript 𝐶 0 superscript subscript 𝑝 1 𝑛 subscript 𝐶 2 𝑝 0 {\displaystyle{\displaystyle\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}>0}}
\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p} > 0		 

(1)/(2)*C[0]+ sum(C[2*p], p = 1..n) > 0
 
Divide[1,2]*Subscript[C, 0]+ Sum[Subscript[C, 2*p], {p, 1, n}, GenerateConditions->None] > 0
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29.15.E25 ∑ p = 0 n B 2 ⁢ p + 2 2 = 1 superscript subscript 𝑝 0 𝑛 superscript subscript 𝐵 2 𝑝 2 2 1 {\displaystyle{\displaystyle\sum_{p=0}^{n}B_{2p+2}^{2}=1}}
\sum_{p=0}^{n}B_{2p+2}^{2} = 1		 

sum((B[2*p + 2])^(2), p = 0..n) = 1
 
Sum[(Subscript[B, 2*p + 2])^(2), {p, 0, n}, GenerateConditions->None] == 1
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29.15.E26 ∑ p = 0 n ( 2 ⁢ p + 2 ) ⁢ B 2 ⁢ p + 2 > 0 superscript subscript 𝑝 0 𝑛 2 𝑝 2 subscript 𝐵 2 𝑝 2 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}(2p+2)B_{2p+2}>0}}
\sum_{p=0}^{n}(2p+2)B_{2p+2} > 0		 

sum((2*p + 2)*B[2*p + 2], p = 0..n) > 0
 
Sum[(2*p + 2)*Subscript[B, 2*p + 2], {p, 0, n}, GenerateConditions->None] > 0
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29.15.E31 ∑ p = 0 n C 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 𝑛 subscript 𝐶 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}C_{2p+1}>0}}
\sum_{p=0}^{n}C_{2p+1} > 0		 

sum(C[2*p + 1], p = 0..n) > 0
 
Sum[Subscript[C, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0
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29.15.E36 ∑ p = 0 n ( 2 ⁢ p + 1 ) ⁢ D 2 ⁢ p + 1 > 0 superscript subscript 𝑝 0 𝑛 2 𝑝 1 subscript 𝐷 2 𝑝 1 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}(2p+1)D_{2p+1}>0}}
\sum_{p=0}^{n}(2p+1)D_{2p+1} > 0		 

sum((2*p + 1)*D[2*p + 1], p = 0..n) > 0
 
Sum[(2*p + 1)*Subscript[D, 2*p + 1], {p, 0, n}, GenerateConditions->None] > 0
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29.15.E40 ( 1 - 1 2 ⁢ k 2 ) ⁢ ∑ p = 0 n D 2 ⁢ p + 2 2 - 1 2 ⁢ k 2 ⁢ ∑ p = 1 n D 2 ⁢ p ⁢ D 2 ⁢ p + 2 = 1 1 1 2 superscript 𝑘 2 superscript subscript 𝑝 0 𝑛 superscript subscript 𝐷 2 𝑝 2 2 1 2 superscript 𝑘 2 superscript subscript 𝑝 1 𝑛 subscript 𝐷 2 𝑝 subscript 𝐷 2 𝑝 2 1 {\displaystyle{\displaystyle\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2% p+2}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{n}D_{2p}D_{2p+2}=1}}
\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{n}D_{2p}D_{2p+2} = 1		 

(1 -(1)/(2)*(k)^(2))*sum((D[2*p + 2])^(2), p = 0..n)-(1)/(2)*(k)^(2)* sum(D[2*p]*D[2*p + 2], p = 1..n) = 1
 
(1 -Divide[1,2]*(k)^(2))*Sum[(Subscript[D, 2*p + 2])^(2), {p, 0, n}, GenerateConditions->None]-Divide[1,2]*(k)^(2)* Sum[Subscript[D, 2*p]*Subscript[D, 2*p + 2], {p, 1, n}, GenerateConditions->None] == 1
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29.15.E41 ∑ p = 0 n ( 2 ⁢ p + 2 ) ⁢ D 2 ⁢ p + 2 > 0 superscript subscript 𝑝 0 𝑛 2 𝑝 2 subscript 𝐷 2 𝑝 2 0 {\displaystyle{\displaystyle\sum_{p=0}^{n}(2p+2)D_{2p+2}>0}}
\sum_{p=0}^{n}(2p+2)D_{2p+2} > 0		 

sum((2*p + 2)*D[2*p + 2], p = 0..n) > 0
 
Sum[(2*p + 2)*Subscript[D, 2*p + 2], {p, 0, n}, GenerateConditions->None] > 0
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