Mathieu Functions and Hill’s Equation - 28.32 Mathematical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.32#Ex1 x = c ⁢ cosh ⁡ ξ ⁢ cos ⁡ η 𝑥 𝑐 𝜉 𝜂 {\displaystyle{\displaystyle x=c\cosh\xi\cos\eta}}
x = c\cosh@@{\xi}\cos@@{\eta}		 

x = c*cosh(xi)*cos(eta)

x == c*Cosh[\[Xi]]*Cos[\[Eta]]
Failure Failure
Failed [300 / 300]
Result: 3.124702180-.2170218424*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = 1/2*3^(1/2)+1/2*I}


Result: 2.064186236-.8699661686*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)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[3.124702180526338, -0.2170218422419914]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.0641862358993213, -0.869966168513175]
Test Values: {Rule[c, -1.5], Rule[x, 1.5], Rule[η, 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
28.32#Ex2 y = c ⁢ sinh ⁡ ξ ⁢ sin ⁡ η 𝑦 𝑐 𝜉 𝜂 {\displaystyle{\displaystyle y=c\sinh\xi\sin\eta}}
y = c\sinh@@{\xi}\sin@@{\eta}		 

y = c*sinh(xi)*sin(eta)

y == c*Sinh[\[Xi]]*Sin[\[Eta]]
Failure Failure
Failed [300 / 300]
Result: -.7333267200+1.299026649*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, y = -3/2}


Result: 2.266673280+1.299026649*I
Test Values: {c = -3/2, eta = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.7333267206780307, 1.2990266484068542]
Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.3699661685131748, 0.9358137641006792]
Test Values: {Rule[c, -1.5], Rule[y, -1.5], Rule[η, 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
28.32.E2 ∂ 2 ⁡ V ∂ ⁡ x 2 + ∂ 2 ⁡ V ∂ ⁡ y 2 + k 2 ⁢ V = 0 partial-derivative 𝑉 𝑥 2 partial-derivative 𝑉 𝑦 2 superscript 𝑘 2 𝑉 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}V}{{\partial x}^{2}}+\frac{{% \partial}^{2}V}{{\partial y}^{2}}+k^{2}V=0}}
\pderiv[2]{V}{x}+\pderiv[2]{V}{y}+k^{2}V = 0		 

diff(V, [x$(2)])+ diff(V, [y$(2)])+ (k)^(2)* V = 0

D[V, {x, 2}]+ D[V, {y, 2}]+ (k)^(2)* V == 0
Failure Failure
Failed [300 / 300]
Result: .8660254040+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 1}


Result: 3.464101616+2.*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[3.464101615137755, 1.9999999999999998]
Test Values: {Rule[k, 2], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}

... skip entries to safe data
28.32.E3 ∂ 2 ⁡ V ∂ ⁡ ξ 2 + ∂ 2 ⁡ V ∂ ⁡ η 2 + 1 2 ⁢ c 2 ⁢ k 2 ⁢ ( cosh ⁡ ( 2 ⁢ ξ ) - cos ⁡ ( 2 ⁢ η ) ) ⁢ V = 0 partial-derivative 𝑉 𝜉 2 partial-derivative 𝑉 𝜂 2 1 2 superscript 𝑐 2 superscript 𝑘 2 2 𝜉 2 𝜂 𝑉 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{% \partial}^{2}V}{{\partial\eta}^{2}}+\frac{1}{2}c^{2}k^{2}(\cosh\left(2\xi% \right)-\cos\left(2\eta\right))V=0}}
\pderiv[2]{V}{\xi}+\pderiv[2]{V}{\eta}+\frac{1}{2}c^{2}k^{2}(\cosh@{2\xi}-\cos@{2\eta})V = 0		 

diff(V, [xi$(2)])+ diff(V, [eta$(2)])+(1)/(2)*(c)^(2)* (k)^(2)*(cosh(2*xi)- cos(2*eta))*V = 0

D[V, {\[Xi], 2}]+ D[V, {\[Eta], 2}]+Divide[1,2]*(c)^(2)* (k)^(2)*(Cosh[2*\[Xi]]- Cos[2*\[Eta]])*V == 0
Failure Failure
Failed [276 / 300]
Result: -.1726552223+4.399682965*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, c = -3/2, eta = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.6906208892+17.59873186*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, c = -3/2, eta = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [276 / 300]
Result: Complex[-0.172655223437435, 4.399682962494039]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.69062089374974, 17.598731849976154]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.32.E4 ∂ 2 ⁡ K ∂ ⁡ z 2 - ∂ 2 ⁡ K ∂ ⁡ ζ 2 = 2 ⁢ q ⁢ ( cos ⁡ ( 2 ⁢ z ) - cos ⁡ ( 2 ⁢ ζ ) ) ⁢ K partial-derivative 𝐾 𝑧 2 partial-derivative 𝐾 𝜁 2 2 𝑞 2 𝑧 2 𝜁 𝐾 {\displaystyle{\displaystyle\frac{{\partial}^{2}K}{{\partial z}^{2}}-\frac{{% \partial}^{2}K}{{\partial\zeta}^{2}}=2q\left(\cos\left(2z\right)-\cos\left(2% \zeta\right)\right)K}}
\pderiv[2]{K}{z}-\pderiv[2]{K}{\zeta} = 2q\left(\cos@{2z}-\cos@{2\zeta}\right)K		 

diff(K, [z$(2)])- diff(K, [zeta$(2)]) = 2*q*(cos(2*z)- cos(2*zeta))*K

D[K, {z, 2}]- D[K, {\[Zeta], 2}] == 2*q*(Cos[2*z]- Cos[2*\[Zeta]])*K
Failure Failure
Failed [240 / 300]
Result: -4.176649406+6.620283744*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}


Result: -4.176649406+6.620283744*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, zeta = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [240 / 300]
Result: Complex[-4.176649405937627, 6.620283737597687]
Test Values: {Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-4.17664940593763, 6.620283737597683]
Test Values: {Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
28.32#Ex3 x 1 = 1 2 ⁢ c ⁢ ( cosh ⁡ ( 2 ⁢ α ) + cos ⁡ ( 2 ⁢ β ) - cosh ⁡ ( 2 ⁢ γ ) ) subscript 𝑥 1 1 2 𝑐 2 𝛼 2 𝛽 2 𝛾 {\displaystyle{\displaystyle x_{1}=\tfrac{1}{2}c\left(\cosh\left(2\alpha\right% )+\cos\left(2\beta\right)-\cosh\left(2\gamma\right)\right)}}
x_{1} = \tfrac{1}{2}c\left(\cosh@{2\alpha}+\cos@{2\beta}-\cosh@{2\gamma}\right)		 

x[1] = (1)/(2)*c*(cosh(2*alpha)+ cos(2*beta)- cosh(2*gamma))

Subscript[x, 1] == Divide[1,2]*c*(Cosh[2*\[Alpha]]+ Cos[2*\[Beta]]- Cosh[2*\[Gamma]])
Failure Failure
Failed [300 / 300]
Result: 6.366481639+.5000000000*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[1] = 1/2*3^(1/2)+1/2*I}


Result: 5.000456235+.8660254040*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[6.493212844498693, -1.2277437153775796]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[5.127187440714255, -0.8617183115931409]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.32#Ex4 x 2 = 2 ⁢ c ⁢ cosh ⁡ α ⁢ cos ⁡ β ⁢ sinh ⁡ γ subscript 𝑥 2 2 𝑐 𝛼 𝛽 𝛾 {\displaystyle{\displaystyle x_{2}=2c\cosh\alpha\cos\beta\sinh\gamma}}
x_{2} = 2c\cosh@@{\alpha}\cos@@{\beta}\sinh@@{\gamma}		 

x[2] = 2*c*cosh(alpha)*cos(beta)*sinh(gamma)

Subscript[x, 2] == 2*c*Cosh[\[Alpha]]*Cos[\[Beta]]*Sinh[\[Gamma]]
Failure Failure
Failed [300 / 300]
Result: 1.170446049+.5000000000*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[2] = 1/2*3^(1/2)+1/2*I}


Result: -.1955793552+.8660254040*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.2946642543328961, 0.8348348760715232]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.07136114945154226, 1.200860279855962]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.32#Ex5 x 3 = 2 ⁢ c ⁢ sinh ⁡ α ⁢ sin ⁡ β ⁢ cosh ⁡ γ subscript 𝑥 3 2 𝑐 𝛼 𝛽 𝛾 {\displaystyle{\displaystyle x_{3}=2c\sinh\alpha\sin\beta\cosh\gamma}}
x_{3} = 2c\sinh@@{\alpha}\sin@@{\beta}\cosh@@{\gamma}		 

x[3] = 2*c*sinh(alpha)*sin(beta)*cosh(gamma)

Subscript[x, 3] == 2*c*Sinh[\[Alpha]]*Sin[\[Beta]]*Cosh[\[Gamma]]
Failure Failure
Failed [300 / 300]
Result: 8.329140826+.5000000000*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[3] = 1/2*3^(1/2)+1/2*I}


Result: 6.963115422+.8660254040*I
Test Values: {alpha = 3/2, beta = 3/2, c = -3/2, gamma = 1/2*3^(1/2)+1/2*I, x[3] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[8.689146837902154, 3.488871718498607]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[7.323121434117715, 3.8548971222830457]
Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[β, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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