Mathieu Functions and Hill’s Equation - 28.31 Equations of Whittaker–Hill and Ince

From testwiki
Revision as of 12:08, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.31#Ex1 ξ 2 = - 4 ⁢ k 2 ⁢ c 2 superscript 𝜉 2 4 superscript 𝑘 2 superscript 𝑐 2 {\displaystyle{\displaystyle\xi^{2}=-4k^{2}c^{2}}}
\xi^{2} = -4k^{2}c^{2}		 

(xi)^(2) = - 4*(k)^(2)* (c)^(2)
 
\[Xi]^(2) == - 4*(k)^(2)* (c)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex2 A = η - 1 8 ⁢ ξ 2 𝐴 𝜂 1 8 superscript 𝜉 2 {\displaystyle{\displaystyle A=\eta-\tfrac{1}{8}\xi^{2}}}
A = \eta-\tfrac{1}{8}\xi^{2}		 

A = eta -(1)/(8)*(xi)^(2)
 
A == \[Eta]-Divide[1,8]*\[Xi]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex3 B = - ( p + 1 ) ⁢ ξ 𝐵 𝑝 1 𝜉 {\displaystyle{\displaystyle B=-(p+1)\xi}}
B = -(p+1)\xi		 

B = -(p + 1)*xi
 
B == -(p + 1)*\[Xi]
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex4 W ⁢ ( z ) = w ⁢ ( z ) ⁢ exp ⁡ ( - 1 4 ⁢ ξ ⁢ cos ⁡ ( 2 ⁢ z ) ) 𝑊 𝑧 𝑤 𝑧 1 4 𝜉 2 𝑧 {\displaystyle{\displaystyle W(z)=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z% \right)\right)}}
W(z) = w(z)\exp@{-\tfrac{1}{4}\xi\cos@{2z}}		 

W(z) = w(z)* exp(-(1)/(4)*xi*cos(2*z))

W[z] == w[z]* Exp[-Divide[1,4]*\[Xi]*Cos[2*z]]
Failure Failure
Failed [300 / 300]
Result: .2817275679-.201842736e-1*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.5394015055-.3903737220*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.2817275677812313, -0.02018427332482242]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.06489049435577782, 0.2500000224743827]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.31.E4 w e , s ⁢ ( z ) = ∑ ℓ = 0 ∞ A 2 ⁢ ℓ + s ⁢ cos ⁡ ( 2 ⁢ ℓ + s ) ⁢ z subscript 𝑤 𝑒 𝑠 𝑧 superscript subscript ℓ 0 subscript 𝐴 2 ℓ 𝑠 2 ℓ 𝑠 𝑧 {\displaystyle{\displaystyle w_{\mathit{e},s}(z)=\sum_{\ell=0}^{\infty}A_{2% \ell+s}\cos(2\ell+s)z}}
w_{\mathit{e},s}(z) = \sum_{\ell=0}^{\infty}A_{2\ell+s}\cos@@{(2\ell+s)z}		 

w[e , s](z) = sum(A[2*ell + s]*cos((2*ell + s)*z), ell = 0..infinity)

Subscript[w, e , s][z] == Sum[Subscript[A, 2*\[ScriptL]+ s]*Cos[(2*\[ScriptL]+ s)*z], {\[ScriptL], 0, Infinity}, GenerateConditions->None]
Error Failure - Skip - No test values generated
28.31.E5 w o , s ⁢ ( z ) = ∑ ℓ = 0 ∞ B 2 ⁢ ℓ + s ⁢ sin ⁡ ( 2 ⁢ ℓ + s ) ⁢ z subscript 𝑤 𝑜 𝑠 𝑧 superscript subscript ℓ 0 subscript 𝐵 2 ℓ 𝑠 2 ℓ 𝑠 𝑧 {\displaystyle{\displaystyle w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2% \ell+s}\sin(2\ell+s)z}}
w_{\mathit{o},s}(z) = \sum_{\ell=0}^{\infty}B_{2\ell+s}\sin@@{(2\ell+s)z}		 

w[o , s](z) = sum(B[2*ell + s]*sin((2*ell + s)*z), ell = 0..infinity)

Subscript[w, o , s][z] == Sum[Subscript[B, 2*\[ScriptL]+ s]*Sin[(2*\[ScriptL]+ s)*z], {\[ScriptL], 0, Infinity}, GenerateConditions->None]
Error Failure - Skip - No test values generated
28.31#Ex5 - 2 ⁢ η ⁢ A 0 + ( 2 + p ) ⁢ ξ ⁢ A 2 = 0 2 𝜂 subscript 𝐴 0 2 𝑝 𝜉 subscript 𝐴 2 0 {\displaystyle{\displaystyle-2\eta A_{0}+(2+p)\xi A_{2}=0}}
-2\eta A_{0}+(2+p)\xi A_{2} = 0		 

- 2*eta*A[0]+(2 + p)*xi*A[2] = 0
 
- 2*\[Eta]*Subscript[A, 0]+(2 + p)*\[Xi]*Subscript[A, 2] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex6 p ⁢ ξ ⁢ A 0 + ( 4 - η ) ⁢ A 2 + ( 1 2 ⁢ p + 2 ) ⁢ ξ ⁢ A 4 = 0 𝑝 𝜉 subscript 𝐴 0 4 𝜂 subscript 𝐴 2 1 2 𝑝 2 𝜉 subscript 𝐴 4 0 {\displaystyle{\displaystyle p\xi A_{0}+(4-\eta)A_{2}+\left(\tfrac{1}{2}p+2% \right)\xi A_{4}=0}}
p\xi A_{0}+(4-\eta)A_{2}+\left(\tfrac{1}{2}p+2\right)\xi A_{4} = 0		 

p*xi*A[0]+(4 - eta)*A[2]+((1)/(2)*p + 2)*xi*A[4] = 0
 
p*\[Xi]*Subscript[A, 0]+(4 - \[Eta])*Subscript[A, 2]+(Divide[1,2]*p + 2)*\[Xi]*Subscript[A, 4] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex7 ( 1 2 ⁢ p - ℓ + 1 ) ⁢ ξ ⁢ A 2 ⁢ ℓ - 2 + ( 4 ⁢ ℓ 2 - η ) ⁢ A 2 ⁢ ℓ + ( 1 2 ⁢ p + ℓ + 1 ) ⁢ ξ ⁢ A 2 ⁢ ℓ + 2 = 0 1 2 𝑝 ℓ 1 𝜉 subscript 𝐴 2 ℓ 2 4 superscript ℓ 2 𝜂 subscript 𝐴 2 ℓ 1 2 𝑝 ℓ 1 𝜉 subscript 𝐴 2 ℓ 2 0 {\displaystyle{\displaystyle(\tfrac{1}{2}p-\ell+1)\xi A_{2\ell-2}+\left(4\ell^% {2}-\eta\right)A_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi A_{2\ell+2}=0}}
(\tfrac{1}{2}p-\ell+1)\xi A_{2\ell-2}+\left(4\ell^{2}-\eta\right)A_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi A_{2\ell+2} = 0		 ℓ ≥ 2 ℓ 2 {\displaystyle{\displaystyle\ell\geq 2}} 
((1)/(2)*p - ell + 1)*xi*A[2*ell - 2]+(4*(ell)^(2)- eta)*A[2*ell]+((1)/(2)*p + ell + 1)*xi*A[2*ell + 2] = 0
 
(Divide[1,2]*p - \[ScriptL]+ 1)*\[Xi]*Subscript[A, 2*\[ScriptL]- 2]+(4*\[ScriptL]^(2)- \[Eta])*Subscript[A, 2*\[ScriptL]]+(Divide[1,2]*p + \[ScriptL]+ 1)*\[Xi]*Subscript[A, 2*\[ScriptL]+ 2] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex8 ( 1 - η + ( 1 2 ⁢ p + 1 2 ) ⁢ ξ ) ⁢ A 1 + ( 1 2 ⁢ p + 3 2 ) ⁢ ξ ⁢ A 3 = 0 1 𝜂 1 2 𝑝 1 2 𝜉 subscript 𝐴 1 1 2 𝑝 3 2 𝜉 subscript 𝐴 3 0 {\displaystyle{\displaystyle\left(1-\eta+\left(\tfrac{1}{2}p+\tfrac{1}{2}% \right)\xi\right)A_{1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi A_{3}=0}}
\left(1-\eta+\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)A_{1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi A_{3} = 0		 

(1 - eta +((1)/(2)*p +(1)/(2))*xi)*A[1]+((1)/(2)*p +(3)/(2))*xi*A[3] = 0
 
(1 - \[Eta]+(Divide[1,2]*p +Divide[1,2])*\[Xi])*Subscript[A, 1]+(Divide[1,2]*p +Divide[3,2])*\[Xi]*Subscript[A, 3] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex9 ( 1 2 ⁢ p - ℓ + 1 2 ) ⁢ ξ ⁢ A 2 ⁢ ℓ - 1 + ( ( 2 ⁢ ℓ + 1 ) 2 - η ) ⁢ A 2 ⁢ ℓ + 1 + ( 1 2 ⁢ p + ℓ + 3 2 ) ⁢ ξ ⁢ A 2 ⁢ ℓ + 3 = 0 1 2 𝑝 ℓ 1 2 𝜉 subscript 𝐴 2 ℓ 1 superscript 2 ℓ 1 2 𝜂 subscript 𝐴 2 ℓ 1 1 2 𝑝 ℓ 3 2 𝜉 subscript 𝐴 2 ℓ 3 0 {\displaystyle{\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi A_{2\ell-1}+% \left((2\ell+1)^{2}-\eta\right)A_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})% \xi A_{2\ell+3}=0}}
(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi A_{2\ell-1}+\left((2\ell+1)^{2}-\eta\right)A_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi A_{2\ell+3} = 0		 ℓ ≥ 1 ℓ 1 {\displaystyle{\displaystyle\ell\geq 1}} 
((1)/(2)*p - ell +(1)/(2))*xi*A[2*ell - 1]+((2*ell + 1)^(2)- eta)*A[2*ell + 1]+((1)/(2)*p + ell +(3)/(2))*xi*A[2*ell + 3] = 0
 
(Divide[1,2]*p - \[ScriptL]+Divide[1,2])*\[Xi]*Subscript[A, 2*\[ScriptL]- 1]+((2*\[ScriptL]+ 1)^(2)- \[Eta])*Subscript[A, 2*\[ScriptL]+ 1]+(Divide[1,2]*p + \[ScriptL]+Divide[3,2])*\[Xi]*Subscript[A, 2*\[ScriptL]+ 3] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex10 ( 1 - η - ( 1 2 ⁢ p + 1 2 ) ⁢ ξ ) ⁢ B 1 + ( 1 2 ⁢ p + 3 2 ) ⁢ ξ ⁢ B 3 = 0 1 𝜂 1 2 𝑝 1 2 𝜉 subscript 𝐵 1 1 2 𝑝 3 2 𝜉 subscript 𝐵 3 0 {\displaystyle{\displaystyle\left(1-\eta-\left(\tfrac{1}{2}p+\tfrac{1}{2}% \right)\xi\right)B_{1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi B_{3}=0}}
\left(1-\eta-\left(\tfrac{1}{2}p+\tfrac{1}{2}\right)\xi\right)B_{1}+\left(\tfrac{1}{2}p+\tfrac{3}{2}\right)\xi B_{3} = 0		 

(1 - eta -((1)/(2)*p +(1)/(2))*xi)*B[1]+((1)/(2)*p +(3)/(2))*xi*B[3] = 0
 
(1 - \[Eta]-(Divide[1,2]*p +Divide[1,2])*\[Xi])*Subscript[B, 1]+(Divide[1,2]*p +Divide[3,2])*\[Xi]*Subscript[B, 3] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex11 ( 1 2 ⁢ p - ℓ + 1 2 ) ⁢ ξ ⁢ B 2 ⁢ ℓ - 1 + ( ( 2 ⁢ ℓ + 1 ) 2 - η ) ⁢ B 2 ⁢ ℓ + 1 + ( 1 2 ⁢ p + ℓ + 3 2 ) ⁢ ξ ⁢ B 2 ⁢ ℓ + 3 = 0 1 2 𝑝 ℓ 1 2 𝜉 subscript 𝐵 2 ℓ 1 superscript 2 ℓ 1 2 𝜂 subscript 𝐵 2 ℓ 1 1 2 𝑝 ℓ 3 2 𝜉 subscript 𝐵 2 ℓ 3 0 {\displaystyle{\displaystyle(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+% \left((2\ell+1)^{2}-\eta\right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})% \xi B_{2\ell+3}=0}}
(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+\left((2\ell+1)^{2}-\eta\right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi B_{2\ell+3} = 0		 ℓ ≥ 1 ℓ 1 {\displaystyle{\displaystyle\ell\geq 1}} 
((1)/(2)*p - ell +(1)/(2))*xi*B[2*ell - 1]+((2*ell + 1)^(2)- eta)*B[2*ell + 1]+((1)/(2)*p + ell +(3)/(2))*xi*B[2*ell + 3] = 0
 
(Divide[1,2]*p - \[ScriptL]+Divide[1,2])*\[Xi]*Subscript[B, 2*\[ScriptL]- 1]+((2*\[ScriptL]+ 1)^(2)- \[Eta])*Subscript[B, 2*\[ScriptL]+ 1]+(Divide[1,2]*p + \[ScriptL]+Divide[3,2])*\[Xi]*Subscript[B, 2*\[ScriptL]+ 3] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex12 ( 4 - η ) ⁢ B 2 + ( 1 2 ⁢ p + 2 ) ⁢ ξ ⁢ B 4 = 0 4 𝜂 subscript 𝐵 2 1 2 𝑝 2 𝜉 subscript 𝐵 4 0 {\displaystyle{\displaystyle(4-\eta)B_{2}+\left(\tfrac{1}{2}p+2\right)\xi B_{4% }=0}}
(4-\eta)B_{2}+\left(\tfrac{1}{2}p+2\right)\xi B_{4} = 0		 

(4 - eta)*B[2]+((1)/(2)*p + 2)*xi*B[4] = 0
 
(4 - \[Eta])*Subscript[B, 2]+(Divide[1,2]*p + 2)*\[Xi]*Subscript[B, 4] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex13 ( 1 2 ⁢ p - ℓ + 1 ) ⁢ ξ ⁢ B 2 ⁢ ℓ - 2 + ( 4 ⁢ ℓ 2 - η ) ⁢ B 2 ⁢ ℓ + ( 1 2 ⁢ p + ℓ + 1 ) ⁢ ξ ⁢ B 2 ⁢ ℓ + 2 = 0 1 2 𝑝 ℓ 1 𝜉 subscript 𝐵 2 ℓ 2 4 superscript ℓ 2 𝜂 subscript 𝐵 2 ℓ 1 2 𝑝 ℓ 1 𝜉 subscript 𝐵 2 ℓ 2 0 {\displaystyle{\displaystyle(\tfrac{1}{2}p-\ell+1)\xi B_{2\ell-2}+(4\ell^{2}-% \eta)B_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi B_{2\ell+2}=0}}
(\tfrac{1}{2}p-\ell+1)\xi B_{2\ell-2}+(4\ell^{2}-\eta)B_{2\ell}+(\tfrac{1}{2}p+\ell+1)\xi B_{2\ell+2} = 0		 ℓ ≥ 2 ℓ 2 {\displaystyle{\displaystyle\ell\geq 2}} 
((1)/(2)*p - ell + 1)*xi*B[2*ell - 2]+(4*(ell)^(2)- eta)*B[2*ell]+((1)/(2)*p + ell + 1)*xi*B[2*ell + 2] = 0
 
(Divide[1,2]*p - \[ScriptL]+ 1)*\[Xi]*Subscript[B, 2*\[ScriptL]- 2]+(4*\[ScriptL]^(2)- \[Eta])*Subscript[B, 2*\[ScriptL]]+(Divide[1,2]*p + \[ScriptL]+ 1)*\[Xi]*Subscript[B, 2*\[ScriptL]+ 2] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.31.E12 1 π ⁢ ∫ 0 2 ⁢ π ( C p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 π ⁢ ∫ 0 2 ⁢ π ( S p m ⁢ ( x , ξ ) ) 2 ⁢ d x 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝐶 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝑆 𝑝 𝑚 𝑥 𝜉 2 𝑥 {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi% )\right)^{2}\mathrm{d}x=\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)% \right)^{2}\mathrm{d}x}}
\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi)\right)^{2}\diff{x} = \dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)\right)^{2}\diff{x}		 

(1)/(Pi)*int(((C[p])^(m)(x , xi))^(2), x = 0..2*Pi) = (1)/(Pi)*int(((S[p])^(m)(x , xi))^(2), x = 0..2*Pi)

Divide[1,Pi]*Integrate[((Subscript[C, p])^(m)[x , \[Xi]])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == Divide[1,Pi]*Integrate[((Subscript[S, p])^(m)[x , \[Xi]])^(2), {x, 0, 2*Pi}, GenerateConditions->None]
Failure Failure Error Error
28.31.E12 1 π ⁢ ∫ 0 2 ⁢ π ( S p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝑆 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi% )\right)^{2}\mathrm{d}x=1}}
\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)\right)^{2}\diff{x} = 1		 

(1)/(Pi)*int(((S[p])^(m)(x , xi))^(2), x = 0..2*Pi) = 1

Divide[1,Pi]*Integrate[((Subscript[S, p])^(m)[x , \[Xi]])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == 1
Failure Failure Error Error
28.31#Ex22 ℎ𝑐 2 ⁢ n 2 ⁢ m ⁢ ( z , - ξ ) = ( - 1 ) m ⁢ ℎ𝑐 2 ⁢ n 2 ⁢ m ⁢ ( 1 2 ⁢ π - z , ξ ) superscript subscript ℎ𝑐 2 𝑛 2 𝑚 𝑧 𝜉 superscript 1 𝑚 superscript subscript ℎ𝑐 2 𝑛 2 𝑚 1 2 𝜋 𝑧 𝜉 {\displaystyle{\displaystyle\mathit{hc}_{2n}^{2m}(z,-\xi)=(-1)^{m}\mathit{hc}_% {2n}^{2m}(\tfrac{1}{2}\pi-z,\xi)}}
\mathit{hc}_{2n}^{2m}(z,-\xi) = (-1)^{m}\mathit{hc}_{2n}^{2m}(\tfrac{1}{2}\pi-z,\xi)		 

(hc[2*n])^(2*m)(z , - xi) = (- 1)^(m)* (hc[2*n])^(2*m)((1)/(2)*Pi - z , xi)
 
(Subscript[hc, 2*n])^(2*m)[z , - \[Xi]] == (- 1)^(m)* (Subscript[hc, 2*n])^(2*m)[Divide[1,2]*Pi - z , \[Xi]]
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex23 ℎ𝑐 2 ⁢ n + 1 2 ⁢ m + 1 ⁢ ( z , - ξ ) = ( - 1 ) m ⁢ ℎ𝑠 2 ⁢ n + 1 2 ⁢ m + 1 ⁢ ( 1 2 ⁢ π - z , ξ ) superscript subscript ℎ𝑐 2 𝑛 1 2 𝑚 1 𝑧 𝜉 superscript 1 𝑚 superscript subscript ℎ𝑠 2 𝑛 1 2 𝑚 1 1 2 𝜋 𝑧 𝜉 {\displaystyle{\displaystyle\mathit{hc}_{2n+1}^{2m+1}(z,-\xi)=(-1)^{m}\mathit{% hs}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)}}
\mathit{hc}_{2n+1}^{2m+1}(z,-\xi) = (-1)^{m}\mathit{hs}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)		 

(hc[2*n + 1])^(2*m + 1)(z , - xi) = (- 1)^(m)* (hs[2*n + 1])^(2*m + 1)((1)/(2)*Pi - z , xi)
 
(Subscript[hc, 2*n + 1])^(2*m + 1)[z , - \[Xi]] == (- 1)^(m)* (Subscript[hs, 2*n + 1])^(2*m + 1)[Divide[1,2]*Pi - z , \[Xi]]
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex24 ℎ𝑠 2 ⁢ n + 1 2 ⁢ m + 1 ⁢ ( z , - ξ ) = ( - 1 ) m ⁢ ℎ𝑐 2 ⁢ n + 1 2 ⁢ m + 1 ⁢ ( 1 2 ⁢ π - z , ξ ) superscript subscript ℎ𝑠 2 𝑛 1 2 𝑚 1 𝑧 𝜉 superscript 1 𝑚 superscript subscript ℎ𝑐 2 𝑛 1 2 𝑚 1 1 2 𝜋 𝑧 𝜉 {\displaystyle{\displaystyle\mathit{hs}_{2n+1}^{2m+1}(z,-\xi)=(-1)^{m}\mathit{% hc}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)}}
\mathit{hs}_{2n+1}^{2m+1}(z,-\xi) = (-1)^{m}\mathit{hc}_{2n+1}^{2m+1}(\tfrac{1}{2}\pi-z,\xi)		 

(hs[2*n + 1])^(2*m + 1)(z , - xi) = (- 1)^(m)* (hc[2*n + 1])^(2*m + 1)((1)/(2)*Pi - z , xi)
 
(Subscript[hs, 2*n + 1])^(2*m + 1)[z , - \[Xi]] == (- 1)^(m)* (Subscript[hc, 2*n + 1])^(2*m + 1)[Divide[1,2]*Pi - z , \[Xi]]
 Skipped - no semantic math Skipped - no semantic math - -
28.31#Ex25 ℎ𝑠 2 ⁢ n + 2 2 ⁢ m + 2 ⁢ ( z , - ξ ) = ( - 1 ) m ⁢ ℎ𝑠 2 ⁢ n + 2 2 ⁢ m + 2 ⁢ ( 1 2 ⁢ π - z , ξ ) superscript subscript ℎ𝑠 2 𝑛 2 2 𝑚 2 𝑧 𝜉 superscript 1 𝑚 superscript subscript ℎ𝑠 2 𝑛 2 2 𝑚 2 1 2 𝜋 𝑧 𝜉 {\displaystyle{\displaystyle\mathit{hs}_{2n+2}^{2m+2}(z,-\xi)=(-1)^{m}\mathit{% hs}_{2n+2}^{2m+2}(\tfrac{1}{2}\pi-z,\xi)}}
\mathit{hs}_{2n+2}^{2m+2}(z,-\xi) = (-1)^{m}\mathit{hs}_{2n+2}^{2m+2}(\tfrac{1}{2}\pi-z,\xi)		 

(hs[2*n + 2])^(2*m + 2)(z , - xi) = (- 1)^(m)* (hs[2*n + 2])^(2*m + 2)((1)/(2)*Pi - z , xi)
 
(Subscript[hs, 2*n + 2])^(2*m + 2)[z , - \[Xi]] == (- 1)^(m)* (Subscript[hs, 2*n + 2])^(2*m + 2)[Divide[1,2]*Pi - z , \[Xi]]
 Skipped - no semantic math Skipped - no semantic math - -
28.31.E21 ∫ 0 2 ⁢ π ℎ𝑐 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑐 p m 2 ⁢ ( x , ξ ) ⁢ d x = ∫ 0 2 ⁢ π ℎ𝑠 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑠 p m 2 ⁢ ( x , ξ ) ⁢ d x superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑐 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)% \mathit{hc}_{p}^{m_{2}}(x,\xi)\mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1% }}(x,\xi)\mathit{hs}_{p}^{m_{2}}(x,\xi)\mathrm{d}x}}
\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)\mathit{hc}_{p}^{m_{2}}(x,\xi)\diff{x} = \int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2}}(x,\xi)\diff{x}		 

int((hc[p])^(m[1])(x , xi)* (hc[p])^(m[2])(x , xi), x = 0..2*Pi) = int((hs[p])^(m[1])(x , xi)* (hs[p])^(m[2])(x , xi), x = 0..2*Pi)

Integrate[(Subscript[hc, p])^(Subscript[m, 1])[x , \[Xi]]* (Subscript[hc, p])^(Subscript[m, 2])[x , \[Xi]], {x, 0, 2*Pi}, GenerateConditions->None] == Integrate[(Subscript[hs, p])^(Subscript[m, 1])[x , \[Xi]]* (Subscript[hs, p])^(Subscript[m, 2])[x , \[Xi]], {x, 0, 2*Pi}, GenerateConditions->None]
Failure Failure Manual Skip! Error
28.31.E21 ∫ 0 2 ⁢ π ℎ𝑠 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑠 p m 2 ⁢ ( x , ξ ) ⁢ d x = 0 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)% \mathit{hs}_{p}^{m_{2}}(x,\xi)\mathrm{d}x=0}}
\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)\mathit{hs}_{p}^{m_{2}}(x,\xi)\diff{x} = 0		 

int((hs[p])^(m[1])(x , xi)* (hs[p])^(m[2])(x , xi), x = 0..2*Pi) = 0

Integrate[(Subscript[hs, p])^(Subscript[m, 1])[x , \[Xi]]* (Subscript[hs, p])^(Subscript[m, 2])[x , \[Xi]], {x, 0, 2*Pi}, GenerateConditions->None] == 0
Failure Failure Manual Skip! Error
28.31.E22 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑐 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑐 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}}
\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos@{2u}-\cos@{2v}\right)\diff{v}\diff{u} = 0		 

int(int((hc[p[1]])^(m[1])(u , xi)* (hc[p[1]])^(m[1])(v , xi)* (hc[p[2]])^(m[2])(u , xi)* (hc[p[2]])^(m[2])(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity]) = 0

Integrate[Integrate[(Subscript[hc, Subscript[p, 1]])^(Subscript[m, 1])[u , \[Xi]]* (Subscript[hc, Subscript[p, 1]])^(Subscript[m, 1])[v , \[Xi]]* (Subscript[hc, Subscript[p, 2]])^(Subscript[m, 2])[u , \[Xi]]* (Subscript[hc, Subscript[p, 2]])^(Subscript[m, 2])[v , \[Xi]]*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}, GenerateConditions->None], {u, Subscript[u, 0], Subscript[u, Infinity]}, GenerateConditions->None] == 0
Failure Failure Error Error
28.31.E23 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑠 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}}
\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs}_{p_{1}}^{m_{1}}(u,\xi)\mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos@{2u}-\cos@{2v}\right)\diff{v}\diff{u} = 0		 

int(int((hs[p[1]])^(m[1])(u , xi)* (hs[p[1]])^(m[1])(v , xi)* (hs[p[2]])^(m[2])(u , xi)* (hs[p[2]])^(m[2])(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity]) = 0

Integrate[Integrate[(Subscript[hs, Subscript[p, 1]])^(Subscript[m, 1])[u , \[Xi]]* (Subscript[hs, Subscript[p, 1]])^(Subscript[m, 1])[v , \[Xi]]* (Subscript[hs, Subscript[p, 2]])^(Subscript[m, 2])[u , \[Xi]]* (Subscript[hs, Subscript[p, 2]])^(Subscript[m, 2])[v , \[Xi]]*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}, GenerateConditions->None], {u, Subscript[u, 0], Subscript[u, Infinity]}, GenerateConditions->None] == 0
Failure Failure Error Error
28.31.E24 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑐 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}}
\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc}_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos@{2u}-\cos@{2v}\right)\diff{v}\diff{u} = 0		 

int(int((hc[p[1]])^(m[1])(u , xi)* (hc[p[1]])^(m[1])(v , xi)* (hs[p[2]])^(m[2])(u , xi)* (hs[p[2]])^(m[2])(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity]) = 0

Integrate[Integrate[(Subscript[hc, Subscript[p, 1]])^(Subscript[m, 1])[u , \[Xi]]* (Subscript[hc, Subscript[p, 1]])^(Subscript[m, 1])[v , \[Xi]]* (Subscript[hs, Subscript[p, 2]])^(Subscript[m, 2])[u , \[Xi]]* (Subscript[hs, Subscript[p, 2]])^(Subscript[m, 2])[v , \[Xi]]*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}, GenerateConditions->None], {u, Subscript[u, 0], Subscript[u, Infinity]}, GenerateConditions->None] == 0
Failure Failure Error Error
Retrieved from "https://lct.wmflabs.org/w/index.php?title=28.31&oldid=58499"