Spheroidal Wave Functions - 30.13 Wave Equation in Prolate Spheroidal Coordinates

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
30.13#Ex4 1 < ΞΎ 1 πœ‰ {\displaystyle{\displaystyle 1<\xi}}
1 < \xi		 

1 < xi
 
1 < \[Xi]
 Skipped - no semantic math Skipped - no semantic math - -
30.13#Ex5 - 1 < Ξ· 1 πœ‚ {\displaystyle{\displaystyle-1<\eta}}
-1 < \eta		 

- 1 < eta
 
- 1 < \[Eta]
 Skipped - no semantic math Skipped - no semantic math - -
30.13#Ex6 0 ≀ Ο• 0 italic-Ο• {\displaystyle{\displaystyle 0\leq\phi}}
0 \leq \phi		 

0 <= phi
 
0 <= \[Phi]
 Skipped - no semantic math Skipped - no semantic math - -
30.13.E3 h ΞΎ 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ ΞΎ ) 2 superscript subscript β„Ž πœ‰ 2 superscript partial-derivative π‘₯ πœ‰ 2 superscript partial-derivative 𝑦 πœ‰ 2 superscript partial-derivative 𝑧 πœ‰ 2 {\displaystyle{\displaystyle h_{\xi}^{2}=\left(\frac{\partial x}{\partial\xi}% \right)^{2}+\left(\frac{\partial y}{\partial\xi}\right)^{2}+\left(\frac{% \partial z}{\partial\xi}\right)^{2}}}
h_{\xi}^{2} = \left(\pderiv{x}{\xi}\right)^{2}+\left(\pderiv{y}{\xi}\right)^{2}+\left(\pderiv{z}{\xi}\right)^{2}		 

(h[xi])^(2) = (diff(x, xi))^(2)+(diff(y, xi))^(2)+(diff(x + y*I, xi))^(2)

(Subscript[h, \[Xi]])^(2) == (D[x, \[Xi]])^(2)+(D[y, \[Xi]])^(2)+(D[x + y*I, \[Xi]])^(2)
Failure Failure
Failed [300 / 300]
Result: .5000000004+.8660254040*I
Test Values: {x = 3/2, xi = 1/2*3^(1/2)+1/2*I, y = -3/2, h[xi] = 1/2*3^(1/2)+1/2*I}


Result: -.5000000004-.8660254040*I
Test Values: {x = 3/2, xi = 1/2*3^(1/2)+1/2*I, y = -3/2, h[xi] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, ΞΎ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4999999999999998, -0.8660254037844387]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, ΞΎ], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E3 ( βˆ‚ ⁑ x βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ ΞΎ ) 2 = c 2 ⁒ ( ΞΎ 2 - Ξ· 2 ) ΞΎ 2 - 1 superscript partial-derivative π‘₯ πœ‰ 2 superscript partial-derivative 𝑦 πœ‰ 2 superscript partial-derivative 𝑧 πœ‰ 2 superscript 𝑐 2 superscript πœ‰ 2 superscript πœ‚ 2 superscript πœ‰ 2 1 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\xi}\right)^{2}+% \left(\frac{\partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{% \partial\xi}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1}}}
\left(\pderiv{x}{\xi}\right)^{2}+\left(\pderiv{y}{\xi}\right)^{2}+\left(\pderiv{z}{\xi}\right)^{2} = \frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1}		 

(diff(x, xi))^(2)+(diff(y, xi))^(2)+(diff(x + y*I, xi))^(2) = ((c)^(2)*((xi)^(2)- (eta)^(2)))/((xi)^(2)- 1)

(D[x, \[Xi]])^(2)+(D[y, \[Xi]])^(2)+(D[x + y*I, \[Xi]])^(2) == Divide[(c)^(2)*(\[Xi]^(2)- \[Eta]^(2)),\[Xi]^(2)- 1]
Failure Failure
Failed [240 / 300]
Result: -2.250000002-1.299038105*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = -1/2+1/2*I*3^(1/2), y = -3/2}


Result: -2.250000002-1.299038105*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = -1/2+1/2*I*3^(1/2), y = 3/2}

... skip entries to safe data
Failed [240 / 300]
Result: Complex[-2.25, -1.2990381056766578]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.25, -1.2990381056766578]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
30.13.E4 h Ξ· 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ξ· ) 2 superscript subscript β„Ž πœ‚ 2 superscript partial-derivative π‘₯ πœ‚ 2 superscript partial-derivative 𝑦 πœ‚ 2 superscript partial-derivative 𝑧 πœ‚ 2 {\displaystyle{\displaystyle h_{\eta}^{2}=\left(\frac{\partial x}{\partial\eta% }\right)^{2}+\left(\frac{\partial y}{\partial\eta}\right)^{2}+\left(\frac{% \partial z}{\partial\eta}\right)^{2}}}
h_{\eta}^{2} = \left(\pderiv{x}{\eta}\right)^{2}+\left(\pderiv{y}{\eta}\right)^{2}+\left(\pderiv{z}{\eta}\right)^{2}		 

(h[eta])^(2) = (diff(x, eta))^(2)+(diff(y, eta))^(2)+(diff(x + y*I, eta))^(2)

(Subscript[h, \[Eta]])^(2) == (D[x, \[Eta]])^(2)+(D[y, \[Eta]])^(2)+(D[x + y*I, \[Eta]])^(2)
Failure Failure
Failed [300 / 300]
Result: .5000000004+.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, h[eta] = 1/2*3^(1/2)+1/2*I}


Result: -.5000000004-.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, h[eta] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, Ξ·], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4999999999999998, -0.8660254037844387]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, Ξ·], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E4 ( βˆ‚ ⁑ x βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ξ· ) 2 = c 2 ⁒ ( ΞΎ 2 - Ξ· 2 ) 1 - Ξ· 2 superscript partial-derivative π‘₯ πœ‚ 2 superscript partial-derivative 𝑦 πœ‚ 2 superscript partial-derivative 𝑧 πœ‚ 2 superscript 𝑐 2 superscript πœ‰ 2 superscript πœ‚ 2 1 superscript πœ‚ 2 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\eta}\right)^{2}+% \left(\frac{\partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{% \partial\eta}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}}}}
\left(\pderiv{x}{\eta}\right)^{2}+\left(\pderiv{y}{\eta}\right)^{2}+\left(\pderiv{z}{\eta}\right)^{2} = \frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}}		 

(diff(x, eta))^(2)+(diff(y, eta))^(2)+(diff(x + y*I, eta))^(2) = ((c)^(2)*((xi)^(2)- (eta)^(2)))/(1 - (eta)^(2))

(D[x, \[Eta]])^(2)+(D[y, \[Eta]])^(2)+(D[x + y*I, \[Eta]])^(2) == Divide[(c)^(2)*(\[Xi]^(2)- \[Eta]^(2)),1 - \[Eta]^(2)]
Failure Failure
Failed [240 / 300]
Result: -2.250000002+3.897114318*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = -1/2+1/2*I*3^(1/2), y = -3/2}


Result: -2.250000002+3.897114318*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = -1/2+1/2*I*3^(1/2), y = 3/2}

... skip entries to safe data
Failed [240 / 300]
Result: Complex[-2.2500000000000004, 3.897114317029973]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.2500000000000004, 3.897114317029973]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
30.13.E5 h Ο• 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ο• ) 2 superscript subscript β„Ž italic-Ο• 2 superscript partial-derivative π‘₯ italic-Ο• 2 superscript partial-derivative 𝑦 italic-Ο• 2 superscript partial-derivative 𝑧 italic-Ο• 2 {\displaystyle{\displaystyle h_{\phi}^{2}=\left(\frac{\partial x}{\partial\phi% }\right)^{2}+\left(\frac{\partial y}{\partial\phi}\right)^{2}+\left(\frac{% \partial z}{\partial\phi}\right)^{2}}}
h_{\phi}^{2} = \left(\pderiv{x}{\phi}\right)^{2}+\left(\pderiv{y}{\phi}\right)^{2}+\left(\pderiv{z}{\phi}\right)^{2}		 

(h[phi])^(2) = (diff(x, phi))^(2)+(diff(y, phi))^(2)+(diff(x + y*I, phi))^(2)

(Subscript[h, \[Phi]])^(2) == (D[x, \[Phi]])^(2)+(D[y, \[Phi]])^(2)+(D[x + y*I, \[Phi]])^(2)
Failure Failure
Failed [300 / 300]
Result: .5000000004+.8660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, h[phi] = 1/2*3^(1/2)+1/2*I}


Result: -.5000000004-.8660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, h[phi] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, Ο•], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4999999999999998, -0.8660254037844387]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, Ο•], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E5 ( βˆ‚ ⁑ x βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ο• ) 2 = c 2 ⁒ ( ΞΎ 2 - 1 ) ⁒ ( 1 - Ξ· 2 ) superscript partial-derivative π‘₯ italic-Ο• 2 superscript partial-derivative 𝑦 italic-Ο• 2 superscript partial-derivative 𝑧 italic-Ο• 2 superscript 𝑐 2 superscript πœ‰ 2 1 1 superscript πœ‚ 2 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\phi}\right)^{2}+% \left(\frac{\partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{% \partial\phi}\right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2})}}
\left(\pderiv{x}{\phi}\right)^{2}+\left(\pderiv{y}{\phi}\right)^{2}+\left(\pderiv{z}{\phi}\right)^{2} = c^{2}(\xi^{2}-1)(1-\eta^{2})		 

(diff(x, phi))^(2)+(diff(y, phi))^(2)+(diff(x + y*I, phi))^(2) = (c)^(2)*((xi)^(2)- 1)*(1 - (eta)^(2))

(D[x, \[Phi]])^(2)+(D[y, \[Phi]])^(2)+(D[x + y*I, \[Phi]])^(2) == (c)^(2)*(\[Xi]^(2)- 1)*(1 - \[Eta]^(2))
Failure Failure
Failed [300 / 300]
Result: -1.125000002-1.948557157*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = 1/2*3^(1/2)+1/2*I, y = -3/2}


Result: -1.125000002-1.948557157*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = 1/2*3^(1/2)+1/2*I, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.125, -1.9485571585149866]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.125, -1.9485571585149866]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο•, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E6 1 h ΞΎ ⁒ h Ξ· ⁒ h Ο• ⁒ ( βˆ‚ βˆ‚ ⁑ ΞΎ ⁑ ( h Ξ· ⁒ h Ο• h ΞΎ ⁒ βˆ‚ βˆ‚ ⁑ ΞΎ ) + βˆ‚ βˆ‚ ⁑ Ξ· ⁑ ( h ΞΎ ⁒ h Ο• h Ξ· ⁒ βˆ‚ βˆ‚ ⁑ Ξ· ) + βˆ‚ βˆ‚ ⁑ Ο• ⁑ ( h ΞΎ ⁒ h Ξ· h Ο• ⁒ βˆ‚ βˆ‚ ⁑ Ο• ) ) = 1 c 2 ⁒ ( ΞΎ 2 - Ξ· 2 ) ⁒ ( βˆ‚ βˆ‚ ⁑ ΞΎ ⁑ ( ( ΞΎ 2 - 1 ) ⁒ βˆ‚ βˆ‚ ⁑ ΞΎ ) + βˆ‚ βˆ‚ ⁑ Ξ· ⁑ ( ( 1 - Ξ· 2 ) ⁒ βˆ‚ βˆ‚ ⁑ Ξ· ) + ΞΎ 2 - Ξ· 2 ( ΞΎ 2 - 1 ) ⁒ ( 1 - Ξ· 2 ) ⁒ βˆ‚ 2 βˆ‚ ⁑ Ο• 2 ) 1 subscript β„Ž πœ‰ subscript β„Ž πœ‚ subscript β„Ž italic-Ο• partial-derivative πœ‰ subscript β„Ž πœ‚ subscript β„Ž italic-Ο• subscript β„Ž πœ‰ partial-derivative πœ‰ partial-derivative πœ‚ subscript β„Ž πœ‰ subscript β„Ž italic-Ο• subscript β„Ž πœ‚ partial-derivative πœ‚ partial-derivative italic-Ο• subscript β„Ž πœ‰ subscript β„Ž πœ‚ subscript β„Ž italic-Ο• partial-derivative italic-Ο• 1 superscript 𝑐 2 superscript πœ‰ 2 superscript πœ‚ 2 partial-derivative πœ‰ superscript πœ‰ 2 1 partial-derivative πœ‰ partial-derivative πœ‚ 1 superscript πœ‚ 2 partial-derivative πœ‚ superscript πœ‰ 2 superscript πœ‚ 2 superscript πœ‰ 2 1 1 superscript πœ‚ 2 partial-derivative italic-Ο• 2 {\displaystyle{\displaystyle\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(\frac{% \partial}{\partial\xi}\left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\frac{\partial}{% \partial\xi}\right)+\frac{\partial}{\partial\eta}\left(\frac{h_{\xi}h_{\phi}}{% h_{\eta}}\frac{\partial}{\partial\eta}\right)+\frac{\partial}{\partial\phi}% \left(\frac{h_{\xi}h_{\eta}}{h_{\phi}}\frac{\partial}{\partial\phi}\right)% \right)=\frac{1}{c^{2}(\xi^{2}-\eta^{2})}\left(\frac{\partial}{\partial\xi}% \left((\xi^{2}-1)\frac{\partial}{\partial\xi}\right)+\frac{\partial}{\partial% \eta}\left((1-\eta^{2})\frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}-\eta% ^{2}}{(\xi^{2}-1)(1-\eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right)}}
\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(\pderiv{}{\xi}\left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\pderiv{}{\xi}\right)+\pderiv{}{\eta}\left(\frac{h_{\xi}h_{\phi}}{h_{\eta}}\pderiv{}{\eta}\right)+\pderiv{}{\phi}\left(\frac{h_{\xi}h_{\eta}}{h_{\phi}}\pderiv{}{\phi}\right)\right) = \frac{1}{c^{2}(\xi^{2}-\eta^{2})}\left(\pderiv{}{\xi}\left((\xi^{2}-1)\pderiv{}{\xi}\right)+\pderiv{}{\eta}\left((1-\eta^{2})\pderiv{}{\eta}\right)+\frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-\eta^{2})}\pderiv[2]{}{\phi}\right)		 

(diff((((xi)^(2)- 1)*diff((1)/((c)^(2)*((xi)^(2)- (eta)^(2)))*((xi)^(2)- 1), xi))+ diff(diff(((1 - (eta)^(2))*diff((diff(((h[eta]*h[phi])/(h[xi])*diff((1)/(h[xi]*h[eta]*h[phi])*(h[eta]*h[phi])/(h[xi]), xi))+ diff(((h[xi]*h[phi])/(h[eta])*diff(((h[eta]*h[phi])/(h[xi])*diff((1)/(h[xi]*h[eta]*h[phi])*(h[eta]*h[phi])/(h[xi]), xi))+ (h[xi]*h[phi])/(h[eta]), eta))+ diff((h[xi]*h[eta])/(h[phi])*diff(((h[xi]*h[phi])/(h[eta])*diff(((h[eta]*h[phi])/(h[xi])*diff((1)/(h[xi]*h[eta]*h[phi])*(h[eta]*h[phi])/(h[xi]), xi))+ (h[xi]*h[phi])/(h[eta]), eta))+ (h[xi]*h[eta])/(h[phi]), phi), phi), eta), xi)) (((xi)^(2)- 1)*diff((1)/((c)^(2)*((xi)^(2)- (eta)^(2)))*((xi)^(2)- 1), xi))+ (1 - (eta)^(2)), eta))+((xi)^(2)- (eta)^(2))/(((xi)^(2)- 1)*(1 - (eta)^(2))), [phi$(2)]), eta), xi))

(D[((\[Xi]^(2)- 1)*D[Divide[1,(c)^(2)*(\[Xi]^(2)- \[Eta]^(2))]*(\[Xi]^(2)- 1), \[Xi]])+ D[D[((1 - \[Eta]^(2))*D[(D[(Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]]*D[Divide[1,Subscript[h, \[Xi]]*Subscript[h, \[Eta]]*Subscript[h, \[Phi]]]*Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]], \[Xi]])+ D[(Divide[Subscript[h, \[Xi]]*Subscript[h, \[Phi]],Subscript[h, \[Eta]]]*D[(Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]]*D[Divide[1,Subscript[h, \[Xi]]*Subscript[h, \[Eta]]*Subscript[h, \[Phi]]]*Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]], \[Xi]])+ Divide[Subscript[h, \[Xi]]*Subscript[h, \[Phi]],Subscript[h, \[Eta]]], \[Eta]])+ D[Divide[Subscript[h, \[Xi]]*Subscript[h, \[Eta]],Subscript[h, \[Phi]]]*D[(Divide[Subscript[h, \[Xi]]*Subscript[h, \[Phi]],Subscript[h, \[Eta]]]*D[(Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]]*D[Divide[1,Subscript[h, \[Xi]]*Subscript[h, \[Eta]]*Subscript[h, \[Phi]]]*Divide[Subscript[h, \[Eta]]*Subscript[h, \[Phi]],Subscript[h, \[Xi]]], \[Xi]])+ Divide[Subscript[h, \[Xi]]*Subscript[h, \[Phi]],Subscript[h, \[Eta]]], \[Eta]])+ Divide[Subscript[h, \[Xi]]*Subscript[h, \[Eta]],Subscript[h, \[Phi]]], \[Phi]], \[Phi]], \[Eta]], \[Xi]]) ((\[Xi]^(2)- 1)*D[Divide[1,(c)^(2)*(\[Xi]^(2)- \[Eta]^(2))]*(\[Xi]^(2)- 1), \[Xi]])+ (1 - \[Eta]^(2)), \[Eta]])+Divide[\[Xi]^(2)- \[Eta]^(2),(\[Xi]^(2)- 1)*(1 - \[Eta]^(2))], {\[Phi], 2}], \[Eta]], \[Xi]])
Translation Error Translation Error - -
30.13.E8 w ⁒ ( ΞΎ , Ξ· , Ο• ) = w 1 ⁒ ( ΞΎ ) ⁒ w 2 ⁒ ( Ξ· ) ⁒ w 3 ⁒ ( Ο• ) 𝑀 πœ‰ πœ‚ italic-Ο• subscript 𝑀 1 πœ‰ subscript 𝑀 2 πœ‚ subscript 𝑀 3 italic-Ο• {\displaystyle{\displaystyle w(\xi,\eta,\phi)=w_{1}(\xi)w_{2}(\eta)w_{3}(\phi)}}
w(\xi,\eta,\phi) = w_{1}(\xi)w_{2}(\eta)w_{3}(\phi)		 

w(xi , eta , phi) = w[1](xi)* w[2](eta)* w[3](phi)
 
w[\[Xi], \[Eta], \[Phi]] == Subscript[w, 1][\[Xi]]* Subscript[w, 2][\[Eta]]* Subscript[w, 3][\[Phi]]
 Skipped - no semantic math Skipped - no semantic math - -
30.13.E9 d d ΞΎ ⁑ ( ( 1 - ΞΎ 2 ) ⁒ d w 1 d ΞΎ ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - ΞΎ 2 ) - ΞΌ 2 1 - ΞΎ 2 ) ⁒ w 1 = 0 derivative πœ‰ 1 superscript πœ‰ 2 derivative subscript 𝑀 1 πœ‰ πœ† superscript 𝛾 2 1 superscript πœ‰ 2 superscript πœ‡ 2 1 superscript πœ‰ 2 subscript 𝑀 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}\xi}\left((1-\xi^{2})% \frac{\mathrm{d}w_{1}}{\mathrm{d}\xi}\right)+\left(\lambda+\gamma^{2}(1-\xi^{2% })-\frac{\mu^{2}}{1-\xi^{2}}\right)w_{1}=0}}
\deriv{}{\xi}\left((1-\xi^{2})\deriv{w_{1}}{\xi}\right)+\left(\lambda+\gamma^{2}(1-\xi^{2})-\frac{\mu^{2}}{1-\xi^{2}}\right)w_{1} = 0		 

diff(((1 - (xi)^(2))*diff(w[1], xi))+(lambda + (gamma)^(2)*(1 - (xi)^(2))-((mu)^(2))/(1 - (xi)^(2)))*w[1], xi) = 0

D[((1 - \[Xi]^(2))*D[Subscript[w, 1], \[Xi]])+(\[Lambda]+ \[Gamma]^(2)*(1 - \[Xi]^(2))-Divide[\[Mu]^(2),1 - \[Xi]^(2)])*Subscript[w, 1], \[Xi]] == 0
Failure Failure
Failed [260 / 300]
Result: .6668220767+1.154969718*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, w[1] = 1/2*3^(1/2)+1/2*I}


Result: -1.154969718+.6668220767*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, w[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.0, 2.220446049250313*^-16]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, 1.9999999999999998]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E10 d d Ξ· ⁑ ( ( 1 - Ξ· 2 ) ⁒ d w 2 d Ξ· ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - Ξ· 2 ) - ΞΌ 2 1 - Ξ· 2 ) ⁒ w 2 = 0 derivative πœ‚ 1 superscript πœ‚ 2 derivative subscript 𝑀 2 πœ‚ πœ† superscript 𝛾 2 1 superscript πœ‚ 2 superscript πœ‡ 2 1 superscript πœ‚ 2 subscript 𝑀 2 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}\eta}\left((1-\eta^{2}% )\frac{\mathrm{d}w_{2}}{\mathrm{d}\eta}\right)+\left(\lambda+\gamma^{2}(1-\eta% ^{2})-\frac{\mu^{2}}{1-\eta^{2}}\right)w_{2}=0}}
\deriv{}{\eta}\left((1-\eta^{2})\deriv{w_{2}}{\eta}\right)+\left(\lambda+\gamma^{2}(1-\eta^{2})-\frac{\mu^{2}}{1-\eta^{2}}\right)w_{2} = 0		 

diff(((1 - (eta)^(2))*diff(w[2], eta))+(lambda + (gamma)^(2)*(1 - (eta)^(2))-((mu)^(2))/(1 - (eta)^(2)))*w[2], eta) = 0

D[((1 - \[Eta]^(2))*D[Subscript[w, 2], \[Eta]])+(\[Lambda]+ \[Gamma]^(2)*(1 - \[Eta]^(2))-Divide[\[Mu]^(2),1 - \[Eta]^(2)])*Subscript[w, 2], \[Eta]] == 0
Failure Failure
Failed [300 / 300]
Result: .6668220767+1.154969718*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I}


Result: -1.154969718+.6668220767*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.0, 2.220446049250313*^-16]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, 1.9999999999999998]
Test Values: {Rule[Ξ³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E11 d 2 w 3 d Ο• 2 + ΞΌ 2 ⁒ w 3 = 0 derivative subscript 𝑀 3 italic-Ο• 2 superscript πœ‡ 2 subscript 𝑀 3 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w_{3}}{{\mathrm{d}\phi}^{2}}% +\mu^{2}w_{3}=0}}
\deriv[2]{w_{3}}{\phi}+\mu^{2}w_{3} = 0		 

diff(w[3], [phi$(2)])+ (mu)^(2)* w[3] = 0

D[Subscript[w, 3], {\[Phi], 2}]+ \[Mu]^(2)* Subscript[w, 3] == 0
Failure Failure
Failed [300 / 300]
Result: .3233738859e-9+1.000000001*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, w[3] = 1/2*3^(1/2)+1/2*I}


Result: -1.000000001+.3464101616e-9*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, w[3] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.0, 1.0]
Test Values: {Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: -1.0
Test Values: {Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.13.E12 w 3 ⁒ ( Ο• ) = a 3 ⁒ cos ⁑ ( m ⁒ Ο• ) + b 3 ⁒ sin ⁑ ( m ⁒ Ο• ) subscript 𝑀 3 italic-Ο• subscript π‘Ž 3 π‘š italic-Ο• subscript 𝑏 3 π‘š italic-Ο• {\displaystyle{\displaystyle w_{3}(\phi)=a_{3}\cos\left(m\phi\right)+b_{3}\sin% \left(m\phi\right)}}
w_{3}(\phi) = a_{3}\cos@{m\phi}+b_{3}\sin@{m\phi}		 

w[3](phi) = a[3]*cos(m*phi)+ b[3]*sin(m*phi)

Subscript[w, 3][\[Phi]] == Subscript[a, 3]*Cos[m*\[Phi]]+ Subscript[b, 3]*Sin[m*\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -.9062441475+.1226650086*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[3] = 1/2*3^(1/2)+1/2*I, b[3] = 1/2*3^(1/2)+1/2*I, w[3] = 1/2*3^(1/2)+1/2*I, m = 1}


Result: -1.278771435+1.396327873*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[3] = 1/2*3^(1/2)+1/2*I, b[3] = 1/2*3^(1/2)+1/2*I, w[3] = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.9062441474191866, 0.12266500824612203]
Test Values: {Rule[m, 1], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.2787714355239146, 1.3963278722366796]
Test Values: {Rule[m, 2], Rule[Ο•, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
30.13.E14 w 1 ⁒ ( ΞΎ ) = a 1 ⁒ S n m ⁒ ( 1 ) ⁑ ( ΞΎ , Ξ³ ) + b 1 ⁒ S n m ⁒ ( 2 ) ⁑ ( ΞΎ , Ξ³ ) subscript 𝑀 1 πœ‰ subscript π‘Ž 1 radial-spheroidal-wave-S π‘š 1 𝑛 πœ‰ 𝛾 subscript 𝑏 1 radial-spheroidal-wave-S π‘š 2 𝑛 πœ‰ 𝛾 {\displaystyle{\displaystyle w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(\xi,\gamma% \right)+b_{1}S^{m(2)}_{n}\left(\xi,\gamma\right)}}
w_{1}(\xi) = a_{1}\radsphwaveS{m}{1}{n}@{\xi}{\gamma}+b_{1}\radsphwaveS{m}{2}{n}@{\xi}{\gamma}		 

Error

Subscript[w, 1][\[Xi]] == Subscript[a, 1]*SpheroidalS1[n, m, \[Xi], \[Gamma]]+ Subscript[b, 1]*SpheroidalS2[n, m, \[Xi], \[Gamma]]
Missing Macro Error Failure - Skipped - Because timed out
30.13.E15 S n m ⁒ ( 1 ) ⁑ ( ΞΎ 0 , Ξ³ ) = 0 radial-spheroidal-wave-S π‘š 1 𝑛 subscript πœ‰ 0 𝛾 0 {\displaystyle{\displaystyle S^{m(1)}_{n}\left(\xi_{0},\gamma\right)=0}}
\radsphwaveS{m}{1}{n}@{\xi_{0}}{\gamma} = 0		 

Error

SpheroidalS1[n, m, Subscript[\[Xi], 0], \[Gamma]] == 0
Missing Macro Error Failure - Skipped - Because timed out
30.13.E16 w 1 ⁒ ( ΞΎ 1 ) = w 1 ⁒ ( ΞΎ 2 ) subscript 𝑀 1 subscript πœ‰ 1 subscript 𝑀 1 subscript πœ‰ 2 {\displaystyle{\displaystyle w_{1}(\xi_{1})=w_{1}(\xi_{2})}}
w_{1}(\xi_{1}) = w_{1}(\xi_{2})		 

w[1](xi[1]) = w[1](xi[2])
 
Subscript[w, 1][Subscript[\[Xi], 1]] == Subscript[w, 1][Subscript[\[Xi], 2]]
 Skipped - no semantic math Skipped - no semantic math - -
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