Mathieu Functions and Hill’s Equation - 28.28 Integrals, Integral Representations, and Integral Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.28.E1 w = cosh ⁡ z ⁢ cos ⁡ t ⁢ cos ⁡ α + sinh ⁡ z ⁢ sin ⁡ t ⁢ sin ⁡ α 𝑤 𝑧 𝑡 𝛼 𝑧 𝑡 𝛼 {\displaystyle{\displaystyle w=\cosh z\cos t\cos\alpha+\sinh z\sin t\sin\alpha}}
w = \cosh@@{z}\cos@@{t}\cos@@{\alpha}+\sinh@@{z}\sin@@{t}\sin@@{\alpha}		 

w = cosh(z)*cos(t)*cos(alpha)+ sinh(z)*sin(t)*sin(alpha)

w == Cosh[z]*Cos[t]*Cos[\[Alpha]]+ Sinh[z]*Sin[t]*Sin[\[Alpha]]
Failure Failure
Failed [299 / 300]
Result: 1.714222282+1.165028049*I
Test Values: {alpha = 3/2, t = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .5264627339+1.356668447*I
Test Values: {alpha = 3/2, t = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [298 / 300]
Result: Complex[1.7142222818783819, 1.165028048919159]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}


Result: Complex[1.2004296775262544, 0.7916410797173274]
Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}

... skip entries to safe data
28.28.E10 0 < ph ⁡ ( h ⁢ ( cosh ⁡ z + 1 ) ) 0 phase ℎ 𝑧 1 {\displaystyle{\displaystyle 0<\operatorname{ph}\left(h(\cosh z+1)\right)}}
0 < \phase@{h(\cosh@@{z}+ 1)}		 

0 < argument(h*(cosh(z)+ 1))

0 < Arg[h*(Cosh[z]+ 1)]
Failure Failure
Failed [35 / 70]
Result: 0. < -.8396703302
Test Values: {h = 1/2-1/2*I*3^(1/2), z = 1/2*3^(1/2)+1/2*I}


Result: 0. < -1.272675688
Test Values: {h = 1/2-1/2*I*3^(1/2), z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [35 / 70]
Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.28.E10 0 < ph ⁡ ( h ⁢ ( cosh ⁡ z - 1 ) ) 0 phase ℎ 𝑧 1 {\displaystyle{\displaystyle 0<\operatorname{ph}\left(h(\cosh z-1)\right)}}
0 < \phase@{h(\cosh@@{z}- 1)}		 

0 < argument(h*(cosh(z)- 1))

0 < Arg[h*(Cosh[z]- 1)]
Failure Failure
Failed [35 / 70]
Result: 0. < -1.643566335
Test Values: {h = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: 0. < -1.643566335
Test Values: {h = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [35 / 70]
Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: False
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
28.28.E10 ph ⁡ ( h ⁢ ( cosh ⁡ z + 1 ) ) < π phase ℎ 𝑧 1 𝜋 {\displaystyle{\displaystyle\operatorname{ph}\left(h(\cosh z+1)\right)<\pi}}
\phase@{h(\cosh@@{z}+ 1)} < \pi		 

argument(h*(cosh(z)+ 1)) < Pi

Arg[h*(Cosh[z]+ 1)] < Pi
Failure Failure
Failed [9 / 70]
Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 3/2}


Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 1/2}

... skip entries to safe data
Failed [9 / 70]
Result: False
Test Values: {Rule[h, -1.5], Rule[z, 1.5]}


Result: False
Test Values: {Rule[h, -1.5], Rule[z, 0.5]}

... skip entries to safe data
28.28.E10 ph ⁡ ( h ⁢ ( cosh ⁡ z - 1 ) ) < π phase ℎ 𝑧 1 𝜋 {\displaystyle{\displaystyle\operatorname{ph}\left(h(\cosh z-1)\right)<\pi}}
\phase@{h(\cosh@@{z}- 1)} < \pi		 

argument(h*(cosh(z)- 1)) < Pi

Arg[h*(Cosh[z]- 1)] < Pi
Failure Failure
Failed [9 / 70]
Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 3/2}


Result: 3.141592654 < 3.141592654
Test Values: {h = -3/2, z = 1/2}

... skip entries to safe data
Failed [9 / 70]
Result: False
Test Values: {Rule[h, -1.5], Rule[z, 1.5]}


Result: False
Test Values: {Rule[h, -1.5], Rule[z, 0.5]}

... skip entries to safe data
28.28#Ex4 R ⁢ ( z , t ) = ( 1 2 ⁢ ( cosh ⁡ ( 2 ⁢ z ) + cos ⁡ ( 2 ⁢ t ) ) ) 1 / 2 𝑅 𝑧 𝑡 superscript 1 2 2 𝑧 2 𝑡 1 2 {\displaystyle{\displaystyle R(z,t)=\left(\tfrac{1}{2}(\cosh\left(2z\right)+% \cos\left(2t\right))\right)^{\ifrac{1}{2}}}}
R(z,t) = \left(\tfrac{1}{2}(\cosh@{2z}+\cos@{2t})\right)^{\ifrac{1}{2}}		 

R(z , t) = ((1)/(2)*(cosh(2*z)+ cos(2*t)))^((1)/(2))

R[z , t] == (Divide[1,2]*(Cosh[2*z]+ Cos[2*t]))^(Divide[1,2])
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, -1.500000000)-.8604472605-.6693200135*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}


Result: (.8660254040+.5000000000*I)*(-.5000000000+.8660254040*I, -1.500000000)-.3385916178+.8564557052*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
28.28#Ex5 R ⁢ ( z , 0 ) = cosh ⁡ z 𝑅 𝑧 0 𝑧 {\displaystyle{\displaystyle R(z,0)=\cosh z}}
R(z,0) = \cosh@@{z}		 

R(z , 0) = cosh(z)

R[z , 0] == Cosh[z]
Failure Failure
Failed [70 / 70]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, 0.)-1.227765517-.4690753764*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: (.8660254040+.5000000000*I)*(-.5000000000+.8660254040*I, 0.)-.7305430189+.3969495503*I
Test Values: {R = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
28.28#Ex6 e 2 ⁢ i ⁢ ϕ = cosh ⁡ ( z + i ⁢ t ) cosh ⁡ ( z - i ⁢ t ) superscript 𝑒 2 imaginary-unit italic-ϕ 𝑧 imaginary-unit 𝑡 𝑧 imaginary-unit 𝑡 {\displaystyle{\displaystyle e^{2\mathrm{i}\phi}=\dfrac{\cosh\left(z+\mathrm{i% }t\right)}{\cosh\left(z-\mathrm{i}t\right)}}}
e^{2\iunit\phi} = \dfrac{\cosh@{z+\iunit t}}{\cosh@{z-\iunit t}}		 

exp(2*I*phi) = (cosh(z + I*t))/(cosh(z - I*t))

Exp[2*I*\[Phi]] == Divide[Cosh[z + I*t],Cosh[z - I*t]]
Failure Failure
Failed [300 / 300]
Result: .9781641542+.5339822543*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, t = -3/2, z = 1/2*3^(1/2)+1/2*I}


Result: 1.021212458+.2569827752*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, t = -3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.978164154574313, 0.5339822543847044]
Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1328205399920523, 0.022001382090719362]
Test Values: {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.28#Ex7 ϕ ⁢ ( z , 0 ) = 0 italic-ϕ 𝑧 0 0 {\displaystyle{\displaystyle\phi(z,0)=0}}
\phi(z,0) = 0		 

phi(z , 0) = 0
 
\[Phi][z , 0] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.28.E28 α ν , m ( 1 ) = 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ me ν ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) ⁢ d t subscript superscript 𝛼 1 𝜈 𝑚 1 2 𝜋 superscript subscript 0 2 𝜋 𝑡 Mathieu-me 𝜈 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 𝑡 {\displaystyle{\displaystyle\alpha^{(1)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi% }\sin t\mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{% 2}\right)\mathrm{d}t}}
\alpha^{(1)}_{\nu,m} = \dfrac{1}{2\pi}\int_{0}^{2\pi}\sin@@{t}\Mathieume{\nu}@{t}{h^{2}}\Mathieume{-\nu-2m-1}@{t}{h^{2}}\diff{t}		 

Error

(Subscript[\[Alpha], \[Nu], m])^(1) == Divide[1,2*Pi]*Integrate[Sin[t]*Sqrt[2]*MathieuC[\[Nu], (h)^(2), t]*Sqrt[2]*MathieuC[- \[Nu]- 2*m - 1, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None]
Missing Macro Error Failure - Skipped - Because timed out
28.28.E41 cosh ⁡ z π 2 ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ se n ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p + 1 ⁢ i ⁢ h ⁢ β ^ n , m ⁢ Dsc 0 ⁡ ( n , m , z ) 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 𝑡 Mathieu-se 𝑛 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 1 imaginary-unit ℎ subscript ^ 𝛽 𝑛 𝑚 Mathieu-Dsc 0 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t% \mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)}{{\sinh% ^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\beta}_{n,m}% \mathrm{Dsc}_{0}\left(n,m,z\right)}}
\dfrac{\cosh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@@{t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{0}@{n}{m}{z}		 

(cosh(z))/((Pi)^(2))*int((sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p + 1)* I*h*((1)/(2*Pi)*int(sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))

Divide[Cosh[z],(Pi)^(2)]*Integrate[Divide[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p + 1)* I*h*(Divide[1,2*Pi]*Integrate[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])
Missing Macro Error Missing Macro Error - -
28.28.E42 sinh ⁡ z π 2 ⁢ ∫ 0 2 ⁢ π cos ⁡ t ⁢ se n ′ ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p ⁢ i ⁢ h ⁢ β ^ n , m ⁢ Dsc 1 ⁡ ( n , m , z ) 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 𝑡 diffop Mathieu-se 𝑛 1 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 imaginary-unit ℎ subscript ^ 𝛽 𝑛 𝑚 Mathieu-Dsc 1 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t% \mathrm{se}_{n}'\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)}{{% \sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{p}\mathrm{i}h\widehat{\beta}_{n,m}% \mathrm{Dsc}_{1}\left(n,m,z\right)}}
\dfrac{\sinh@@{z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos@@{t}\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit h\widehat{\beta}_{n,m}\radMathieuDsc{1}@{n}{m}{z}		 

(sinh(z))/((Pi)^(2))*int((cos(t)*subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p)* I*h*((1)/(2*Pi)*int(sin(t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))

Divide[Sinh[z],(Pi)^(2)]*Integrate[Divide[Cos[t]*(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p)* I*h*(Divide[1,2*Pi]*Integrate[Sin[t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])
Missing Macro Error Missing Macro Error - -
28.28.E44 1 π 2 ⁢ ∫ 0 2 ⁢ π sin ⁡ ( 2 ⁢ t ) ⁢ se n ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p ⁢ i ⁢ γ ^ n , m ⁢ Dsc 0 ⁡ ( n , m , z ) 1 superscript 𝜋 2 superscript subscript 0 2 𝜋 2 𝑡 Mathieu-se 𝑛 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 imaginary-unit subscript ^ 𝛾 𝑛 𝑚 Mathieu-Dsc 0 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin\left(% 2t\right)\mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right% )}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{p}\mathrm{i}\widehat{\gamma}_{n,% m}\mathrm{Dsc}_{0}\left(n,m,z\right)}}
\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin@{2t}\Mathieuse{n}@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{0}@{n}{m}{z}		 

(1)/((Pi)^(2))*int((sin(2*t)*MathieuSE(n, (h)^(2), t)*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p)* I*((1)/(2*Pi)*int(subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))

Divide[1,(Pi)^(2)]*Integrate[Divide[Sin[2*t]*MathieuS[n, (h)^(2), t]*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p)* I*(Divide[1,2*Pi]*Integrate[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])
Missing Macro Error Missing Macro Error - -
28.28.E45 sinh ⁡ ( 2 ⁢ z ) π 2 ⁢ ∫ 0 2 ⁢ π se n ′ ⁡ ( t , h 2 ) ⁢ ce m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) p + 1 ⁢ i ⁢ γ ^ n , m ⁢ Dsc 1 ⁡ ( n , m , z ) 2 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 diffop Mathieu-se 𝑛 1 𝑡 superscript ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑝 1 imaginary-unit subscript ^ 𝛾 𝑛 𝑚 Mathieu-Dsc 1 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\sinh\left(2z\right)}{\pi^{2}}\int_{0}^{2% \pi}\dfrac{\mathrm{se}_{n}'\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}% \right)}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}\widehat{% \gamma}_{n,m}\mathrm{Dsc}_{1}\left(n,m,z\right)}}
\dfrac{\sinh@{2z}}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\Mathieuse{n}'@{t}{h^{2}}\Mathieuce{m}@{t}{h^{2}}}{\sinh^{2}@@{z}+\sin^{2}@@{t}}\diff{t} = (-1)^{p+1}\iunit\widehat{\gamma}_{n,m}\radMathieuDsc{1}@{n}{m}{z}		 

(sinh(2*z))/((Pi)^(2))*int((subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t))/((sinh(z))^(2)+ (sin(t))^(2)), t = 0..2*Pi) = (- 1)^(p + 1)* I*((1)/(2*Pi)*int(subs( temp=t, diff( MathieuSE(n, (h)^(2), temp), temp$(1) ) )*MathieuCE(m, (h)^(2), t), t = 0..2*Pi))

Divide[Sinh[2*z],(Pi)^(2)]*Integrate[Divide[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t],(Sinh[z])^(2)+ (Sin[t])^(2)], {t, 0, 2*Pi}, GenerateConditions->None] == (- 1)^(p + 1)* I*(Divide[1,2*Pi]*Integrate[(D[MathieuS[n, (h)^(2), temp], {temp, 1}]/.temp-> t)*MathieuC[m, (h)^(2), t], {t, 0, 2*Pi}, GenerateConditions->None])
Missing Macro Error Missing Macro Error - -
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