Mathieu Functions and Hill’s Equation - 28.5 Second Solutions ,

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.5.E5 ( C n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( f n ⁢ ( x , q ) ) 2 ⁢ d x = ( S n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( g n ⁢ ( x , q ) ) 2 ⁢ d x superscript subscript 𝐶 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑓 𝑛 𝑥 𝑞 2 𝑥 superscript subscript 𝑆 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑔 𝑛 𝑥 𝑞 2 𝑥 {\displaystyle{\displaystyle(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}% \mathrm{d}x=(S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}\mathrm{d}x}}
(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\diff{x} = (S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}\diff{x}		 

(C[n](q))^(2)* int((f[n](x , q))^(2), x = 0..2*Pi) = (S[n](q))^(2)* int((g[n](x , q))^(2), x = 0..2*Pi)

(Subscript[C, n][q])^(2)* Integrate[(Subscript[f, n][x , q])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == (Subscript[S, n][q])^(2)* Integrate[(Subscript[g, n][x , q])^(2), {x, 0, 2*Pi}, GenerateConditions->None]
Failure Failure
Failed [240 / 300]
Result: -165.3668092+.1069227006e-6*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, C[n] = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, g[n] = -1/2+1/2*I*3^(1/2), n = 1}


Result: -165.3668092+.1069227006e-6*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, C[n] = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, g[n] = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
 Error
28.5.E5 ( S n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( g n ⁢ ( x , q ) ) 2 ⁢ d x = π superscript subscript 𝑆 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑔 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle(S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}% \mathrm{d}x=\pi}}
(S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}\diff{x} = \pi		 

(S[n](q))^(2)* int((g[n](x , q))^(2), x = 0..2*Pi) = Pi

(Subscript[S, n][q])^(2)* Integrate[(Subscript[g, n][x , q])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == Pi
Failure Failure
Failed [300 / 300]
Result: -85.82499725+.5347530766e-7*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -85.82499725+.5347530766e-7*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, S[n] = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Error
28.5#Ex1 C 2 ⁢ m ⁢ ( - q ) = C 2 ⁢ m ⁢ ( q ) subscript 𝐶 2 𝑚 𝑞 subscript 𝐶 2 𝑚 𝑞 {\displaystyle{\displaystyle C_{2m}(-q)=C_{2m}(q)}}
C_{2m}(-q) = C_{2m}(q)		 

C[2*m](- q) = C[2*m](q)
 
Subscript[C, 2*m][- q] == Subscript[C, 2*m][q]
 Skipped - no semantic math Skipped - no semantic math - -
28.5#Ex2 C 2 ⁢ m + 1 ⁢ ( - q ) = S 2 ⁢ m + 1 ⁢ ( q ) subscript 𝐶 2 𝑚 1 𝑞 subscript 𝑆 2 𝑚 1 𝑞 {\displaystyle{\displaystyle C_{2m+1}(-q)=S_{2m+1}(q)}}
C_{2m+1}(-q) = S_{2m+1}(q)		 

C[2*m + 1](- q) = S[2*m + 1](q)
 
Subscript[C, 2*m + 1][- q] == Subscript[S, 2*m + 1][q]
 Skipped - no semantic math Skipped - no semantic math - -
28.5#Ex3 S 2 ⁢ m + 2 ⁢ ( - q ) = S 2 ⁢ m + 2 ⁢ ( q ) subscript 𝑆 2 𝑚 2 𝑞 subscript 𝑆 2 𝑚 2 𝑞 {\displaystyle{\displaystyle S_{2m+2}(-q)=S_{2m+2}(q)}}
S_{2m+2}(-q) = S_{2m+2}(q)		 

S[2*m + 2](- q) = S[2*m + 2](q)
 
Subscript[S, 2*m + 2][- q] == Subscript[S, 2*m + 2][q]
 Skipped - no semantic math Skipped - no semantic math - -
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