Mathieu Functions and Hill’s Equation - 28.2 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.2.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z+\pi) = e^{\pi\iunit\nu}w(z)}
w(z+\pi) = e^{\pi\iunit\nu}w(z)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(z + Pi) = exp(Pi*I*nu)*w(z)

w[z + Pi] == Exp[Pi*I*\[Nu]]*w[z]
Failure Failure
Failed [300 / 300]
Result: 3.389122976+2.558671223*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: 1.732824151+2.239220255*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 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[3.3891229743891893, 2.5586712226918134]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.163689701656905, 2.469736091084983]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.2.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z+\pi)+w(z-\pi) = 2\cos@{\pi\nu}w(z)}
w(z+\pi)+w(z-\pi) = 2\cos@{\pi\nu}w(z)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(z + Pi)+ w(z - Pi) = 2*cos(Pi*nu)*w(z)

w[z + Pi]+ w[z - Pi] == 2*Cos[Pi*\[Nu]]*w[z]
Failure Failure
Failed [300 / 300]
Result: 1.661616693+6.639028674*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -6.639028674+1.661616692*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [240 / 300]
Result: Complex[1.6616166873386105, 6.63902867151764]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[14.098728614058, -5.830503683799378]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.2.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z) = \sum_{n=-\infty}^{\infty}c_{2n}e^{\iunit(\nu+2n)z}}
w(z) = \sum_{n=-\infty}^{\infty}c_{2n}e^{\iunit(\nu+2n)z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(z) = sum(c[2*n]*exp(I*(nu + 2*n)*z), n = - infinity..infinity)

w[z] == Sum[Subscript[c, 2*n]*Exp[I*(\[Nu]+ 2*n)*z], {n, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
28.2.E19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2} = 0,}
qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2} = 0,		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
q*c[2*n + 2]-(a -(nu + 2*n)^(2))*c[2*n]+ q*c[2*n - 2] = 0
 
q*Subscript[c, 2*n + 2]-(a -(\[Nu]+ 2*n)^(2))*Subscript[c, 2*n]+ q*Subscript[c, 2*n - 2] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.2.E20 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to+\infty}|c_{2n}|^{1/|n|} = 0}
\lim_{n\to+\infty}|c_{2n}|^{1/|n|} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
limit((abs(c[2*n]))^(1/abs(n)), n = + infinity) = 0
 
Limit[(Abs[Subscript[c, 2*n]])^(1/Abs[n]), n -> + Infinity, GenerateConditions->None] == 0
 Skipped - no semantic math Skipped - no semantic math - -
28.2.E23 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieueigvala{n}@{0} = n^{2}}
\Mathieueigvala{n}@{0} = n^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuA(n, 0) = (n)^(2)

MathieuCharacteristicA[n, 0] == (n)^(2)
Successful Successful - Successful [Tested: 1]
28.2.E24 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieueigvalb{n}@{0} = n^{2}}
\Mathieueigvalb{n}@{0} = n^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuB(n, 0) = (n)^(2)

MathieuCharacteristicB[n, 0] == (n)^(2)
Successful Successful - Successful [Tested: 1]
28.2.E26 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieueigvala{2n}@{-q} = \Mathieueigvala{2n}@{q}}
\Mathieueigvala{2n}@{-q} = \Mathieueigvala{2n}@{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuA(2*n, - q) = MathieuA(2*n, q)

MathieuCharacteristicA[2*n, - q] == MathieuCharacteristicA[2*n, q]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
28.2.E27 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieueigvala{2n+1}@{-q} = \Mathieueigvalb{2n+1}@{q}}
\Mathieueigvala{2n+1}@{-q} = \Mathieueigvalb{2n+1}@{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuA(2*n + 1, - q) = MathieuB(2*n + 1, q)

MathieuCharacteristicA[2*n + 1, - q] == MathieuCharacteristicB[2*n + 1, q]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
28.2.E28 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieueigvalb{2n+2}@{-q} = \Mathieueigvalb{2n+2}@{q}}
\Mathieueigvalb{2n+2}@{-q} = \Mathieueigvalb{2n+2}@{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuB(2*n + 2, - q) = MathieuB(2*n + 2, q)

MathieuCharacteristicB[2*n + 2, - q] == MathieuCharacteristicB[2*n + 2, q]
Failure Failure Successful [Tested: 30] Successful [Tested: 30]
28.2#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuce{0}@{z}{0} = 1/\sqrt{2}}
\Mathieuce{0}@{z}{0} = 1/\sqrt{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuCE(0, 0, z) = 1/(sqrt(2))

MathieuC[0, 0, z] == 1/(Sqrt[2])
Failure Successful Skip - No test values generated Successful [Tested: 7]
28.2#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuce{n}@{z}{0} = \cos@{nz}}
\Mathieuce{n}@{z}{0} = \cos@{nz}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuCE(n, 0, z) = cos(n*z)

MathieuC[n, 0, z] == Cos[n*z]
Successful Failure -
Failed [14 / 21]
Result: Complex[0.6753267742469401, 0.4379310296367226]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.1123802552186532, 0.12519411502047795]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuse{n}@{z}{0} = \sin@{nz}}
\Mathieuse{n}@{z}{0} = \sin@{nz}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuSE(n, 0, z) = sin(n*z)

MathieuS[n, 0, z] == Sin[n*z]
Successful Failure -
Failed [7 / 7]
Result: Complex[0.17898073764673827, 1.8916506821927568]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[4.947243351054952, 0.9068272427732345]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.2#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{2\pi}\left(\Mathieuce{n}@{x}{q}\right)^{2}\diff{x} = \pi}
\int_{0}^{2\pi}\left(\Mathieuce{n}@{x}{q}\right)^{2}\diff{x} = \pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((MathieuCE(n, q, x))^(2), x = 0..2*Pi) = Pi

Integrate[(MathieuC[n, q, x])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == Pi
Failure Failure Skipped - Because timed out
Failed [30 / 30]
Result: Complex[6.9214963829238805, 34.195194735367046]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.5092269783308243, -0.4627812517943034]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{2\pi}\left(\Mathieuse{n}@{x}{q}\right)^{2}\diff{x} = \pi}
\int_{0}^{2\pi}\left(\Mathieuse{n}@{x}{q}\right)^{2}\diff{x} = \pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((MathieuSE(n, q, x))^(2), x = 0..2*Pi) = Pi

Integrate[(MathieuS[n, q, x])^(2), {x, 0, 2*Pi}, GenerateConditions->None] == Pi
Failure Failure
Failed [12 / 30]
Result: -.15495486e-1+.3109277201e-1*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -1.592260336+2.720760990*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-11.13627493115099, -34.66471446201499]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-4.303849824281496, -4.82944497847242]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2.E31 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuce{n}@{x}{q}\diff{x} = 0}
\int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuce{n}@{x}{q}\diff{x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \neq m} 
int(MathieuCE(m, q, x)*MathieuCE(n, q, x), x = 0..2*Pi) = 0

Integrate[MathieuC[m, q, x]*MathieuC[n, q, x], {x, 0, 2*Pi}, GenerateConditions->None] == 0
Failure Failure Skipped - Because timed out Skipped - Because timed out
28.2.E32 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{2\pi}\Mathieuse{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0}
\int_{0}^{2\pi}\Mathieuse{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \neq m} 
int(MathieuSE(m, q, x)*MathieuSE(n, q, x), x = 0..2*Pi) = 0

Integrate[MathieuS[m, q, x]*MathieuS[n, q, x], {x, 0, 2*Pi}, GenerateConditions->None] == 0
Failure Failure Skipped - Because timed out Skipped - Because timed out
28.2.E33 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0}
\int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(MathieuCE(m, q, x)*MathieuSE(n, q, x), x = 0..2*Pi) = 0

Integrate[MathieuC[m, q, x]*MathieuS[n, q, x], {x, 0, 2*Pi}, GenerateConditions->None] == 0
Failure Failure Skipped - Because timed out Skipped - Because timed out
28.2.E34 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuce{2n}@{z}{-q} = (-1)^{n}\Mathieuce{2n}@{\tfrac{1}{2}\pi-z}{q}}
\Mathieuce{2n}@{z}{-q} = (-1)^{n}\Mathieuce{2n}@{\tfrac{1}{2}\pi-z}{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuCE(2*n, - q, z) = (- 1)^(n)* MathieuCE(2*n, q, (1)/(2)*Pi - z)

MathieuC[2*n, - q, z] == (- 1)^(n)* MathieuC[2*n, q, Divide[1,2]*Pi - z]
Failure Failure Successful [Tested: 210]
Failed [210 / 210]
Result: Complex[-0.40308591506050084, 0.46785287118948815]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.60084404002985, 1.182666432116677]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2.E35 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuce{2n+1}@{z}{-q} = (-1)^{n}\Mathieuse{2n+1}@{\tfrac{1}{2}\pi-z}{q}}
\Mathieuce{2n+1}@{z}{-q} = (-1)^{n}\Mathieuse{2n+1}@{\tfrac{1}{2}\pi-z}{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuCE(2*n + 1, - q, z) = (- 1)^(n)* MathieuSE(2*n + 1, q, (1)/(2)*Pi - z)

MathieuC[2*n + 1, - q, z] == (- 1)^(n)* MathieuS[2*n + 1, q, Divide[1,2]*Pi - z]
Failure Failure Successful [Tested: 210]
Failed [210 / 210]
Result: Complex[1.5024747894079764, -2.6392504264802374]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.189026591129222, 0.3274807845663039]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2.E36 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuse{2n+1}@{z}{-q} = (-1)^{n}\Mathieuce{2n+1}@{\tfrac{1}{2}\pi-z}{q}}
\Mathieuse{2n+1}@{z}{-q} = (-1)^{n}\Mathieuce{2n+1}@{\tfrac{1}{2}\pi-z}{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuSE(2*n + 1, - q, z) = (- 1)^(n)* MathieuCE(2*n + 1, q, (1)/(2)*Pi - z)

MathieuS[2*n + 1, - q, z] == (- 1)^(n)* MathieuC[2*n + 1, q, Divide[1,2]*Pi - z]
Failure Failure Successful [Tested: 210]
Failed [210 / 210]
Result: Complex[0.280260494012772, -3.1853558239364403]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.634104542197209, -1.1703184896606507]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.2.E37 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Mathieuse{2n+2}@{z}{-q} = (-1)^{n}\Mathieuse{2n+2}@{\tfrac{1}{2}\pi-z}{q}}
\Mathieuse{2n+2}@{z}{-q} = (-1)^{n}\Mathieuse{2n+2}@{\tfrac{1}{2}\pi-z}{q}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
MathieuSE(2*n + 2, - q, z) = (- 1)^(n)* MathieuSE(2*n + 2, q, (1)/(2)*Pi - z)

MathieuS[2*n + 2, - q, z] == (- 1)^(n)* MathieuS[2*n + 2, q, Divide[1,2]*Pi - z]
Failure Failure
Failed [210 / 210]
Result: -.3430671662+7.821986266*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 20.99712460-1.294028748*I
Test Values: {q = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[4.02456715747845, -1.021331524922309]
Test Values: {Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.169415024309792, -3.4466753320968735]
Test Values: {Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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