Mathieu Functions and Hill’s Equation - 28.29 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.29.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle Q(z+\pi) = Q(z)}
Q(z+\pi) = Q(z)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Q(z + Pi) = Q(z)
 
Q[z + Pi] == Q[z]
 Skipped - no semantic math Skipped - no semantic math - -
28.29.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}Q(z)\diff{z} = 0}
\int_{0}^{\pi}Q(z)\diff{z} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(Q(z), z = 0..Pi) = 0

Integrate[Q[z], {z, 0, Pi}, GenerateConditions->None] == 0
Failure Failure
Failed [10 / 10]
Result: 4.273664071+2.467401101*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I}


Result: -2.467401101+4.273664071*I
Test Values: {Q = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[4.2736640683230425, 2.467401100272339]
Test Values: {Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.4674011002723386, 4.2736640683230425]
Test Values: {Rule[Q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
28.29.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 < \realpart@@{\nu}}
-1 < \realpart@@{\nu}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
- 1 < Re(nu)

- 1 < Re[\[Nu]]
Failure Failure
Failed [2 / 10]
Result: -1. < -1.500000000
Test Values: {nu = -3/2}


Result: -1. < -2.
Test Values: {nu = -2}

Failed [2 / 10]
Result: False
Test Values: {Rule[ν, -1.5]}


Result: False
Test Values: {Rule[ν, -2]}

28.29.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\nu} \leq 1}
\realpart@@{\nu} \leq 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Re(nu) <= 1

Re[\[Nu]] <= 1
Failure Failure
Failed [2 / 10]
Result: 1.500000000 <= 1.
Test Values: {nu = 3/2}


Result: 2. <= 1.
Test Values: {nu = 2}

Failed [2 / 10]
Result: False
Test Values: {Rule[ν, 1.5]}


Result: False
Test Values: {Rule[ν, 2]}

28.29.E7 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
28.29.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z+\pi) = (-1)^{\nu}w(z)+cP(z)}
w(z+\pi) = (-1)^{\nu}w(z)+cP(z)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(z + Pi) = (- 1)^(nu)* w(z)+ cP(z)
 
w[z + Pi] == (- 1)^\[Nu]* w[z]+ cP[z]
 Skipped - no semantic math Skipped - no semantic math - -
28.29.E13 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
28.29.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda_{0} < \mu_{1}}
\lambda_{0} < \mu_{1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
lambda[0](<)*mu[1]
 
Subscript[\[Lambda], 0][<]*Subscript[\[Mu], 1]
 Skipped - no semantic math Skipped - no semantic math - -
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