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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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]]
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \phase@{h(\cosh@@{z}+ 1)}}
0 < \phase@{h(\cosh@@{z}+ 1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < \phase@{h(\cosh@@{z}- 1)}}
0 < \phase@{h(\cosh@@{z}- 1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phase@{h(\cosh@@{z}+ 1)} < \pi}
\phase@{h(\cosh@@{z}+ 1)} < \pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phase@{h(\cosh@@{z}- 1)} < \pi}
\phase@{h(\cosh@@{z}- 1)} < \pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle R(z,t) = \left(\tfrac{1}{2}(\cosh@{2z}+\cos@{2t})\right)^{\ifrac{1}{2}}}
R(z,t) = \left(\tfrac{1}{2}(\cosh@{2z}+\cos@{2t})\right)^{\ifrac{1}{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle R(z,0) = \cosh@@{z}}
R(z,0) = \cosh@@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{2\iunit\phi} = \dfrac{\cosh@{z+\iunit t}}{\cosh@{z-\iunit t}}}
e^{2\iunit\phi} = \dfrac{\cosh@{z+\iunit t}}{\cosh@{z-\iunit t}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi(z,0) = 0}
\phi(z,0) = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
phi(z , 0) = 0
 
\[Phi][z , 0] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.28.E28 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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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