Mathieu Functions and Hill’s Equation - 28.12 Definitions and Basic Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
28.12.E4 me ν ⁡ ( z , 0 ) = e i ⁢ ν ⁢ z Mathieu-me 𝜈 𝑧 0 superscript 𝑒 imaginary-unit 𝜈 𝑧 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,0\right)=e^{\mathrm{i}\nu z% }}}
\Mathieume{\nu}@{z}{0} = e^{\iunit\nu z}		 

Error

Sqrt[2]*MathieuC[\[Nu], 0, z] == Exp[I*\[Nu]*z]
Missing Macro Error Failure -
Failed [70 / 70]
Result: Complex[0.9861942690160291, -0.9067989679250835]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.7892990057566478, 0.4620307840711049]
Test Values: {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
28.12.E5 ∫ 0 π me ν ⁡ ( x , q ) ⁢ me ν ⁡ ( - x , q ) ⁢ d x = π superscript subscript 0 𝜋 Mathieu-me 𝜈 𝑥 𝑞 Mathieu-me 𝜈 𝑥 𝑞 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{\pi}\mathrm{me}_{\nu}\left(x,q\right)% \mathrm{me}_{\nu}\left(-x,q\right)\mathrm{d}x=\pi}}
\int_{0}^{\pi}\Mathieume{\nu}@{x}{q}\Mathieume{\nu}@{-x}{q}\diff{x} = \pi		 

Error

Integrate[Sqrt[2]*MathieuC[\[Nu], q, x]*Sqrt[2]*MathieuC[\[Nu], q, - x], {x, 0, Pi}, GenerateConditions->None] == Pi
Missing Macro Error Failure - Skipped - Because timed out
28.12.E6 me ν ⁡ ( z + π , q ) = e π ⁢ i ⁢ ν ⁢ me ν ⁡ ( z , q ) Mathieu-me 𝜈 𝑧 𝜋 𝑞 superscript 𝑒 𝜋 imaginary-unit 𝜈 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z+\pi,q\right)=e^{\pi% \mathrm{i}\nu}\mathrm{me}_{\nu}\left(z,q\right)}}
\Mathieume{\nu}@{z+\pi}{q} = e^{\pi\iunit\nu}\Mathieume{\nu}@{z}{q}		 

Error

Sqrt[2]*MathieuC[\[Nu], q, z + Pi] == Exp[Pi*I*\[Nu]]*Sqrt[2]*MathieuC[\[Nu], q, z]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-6.370347292395534, -6.192387567232969]
Test Values: {Rule[q, 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[33.312348543319324, -34.35988503520594]
Test Values: {Rule[q, 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
28.12.E7 ∫ 0 π me ν + 2 ⁢ m ⁡ ( x , q ) ⁢ me ν + 2 ⁢ n ⁡ ( - x , q ) ⁢ d x = 0 , superscript subscript 0 𝜋 Mathieu-me 𝜈 2 𝑚 𝑥 𝑞 Mathieu-me 𝜈 2 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{\pi}\mathrm{me}_{\nu+2m}\left(x,q\right)% \mathrm{me}_{\nu+2n}\left(-x,q\right)\mathrm{d}x=0,}}
\int_{0}^{\pi}\Mathieume{\nu+2m}@{x}{q}\Mathieume{\nu+2n}@{-x}{q}\diff{x} = 0,		 m ≠ n 𝑚 𝑛 {\displaystyle{\displaystyle m\neq n}} 
Error

Integrate[Sqrt[2]*MathieuC[\[Nu]+ 2*m, q, x]*Sqrt[2]*MathieuC[\[Nu]+ 2*n, q, - x], {x, 0, Pi}, GenerateConditions->None] == 0
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
28.12.E8 me - ν ⁡ ( z , q ) = me ν ⁡ ( - z , q ) Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{-\nu}\left(z,q\right)=\mathrm{me}_{% \nu}\left(-z,q\right)}}
\Mathieume{-\nu}@{z}{q} = \Mathieume{\nu}@{-z}{q}		 

Error

Sqrt[2]*MathieuC[- \[Nu], q, z] == Sqrt[2]*MathieuC[\[Nu], q, - z]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-3.065655571425399, 0.7817797951498487]
Test Values: {Rule[q, 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[-0.2535777606795988, -2.2365806414914347]
Test Values: {Rule[q, 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
28.12.E9 me ν ⁡ ( z , - q ) = e i ⁢ ν ⁢ π / 2 ⁢ me ν ⁡ ( z - 1 2 ⁢ π , q ) Mathieu-me 𝜈 𝑧 𝑞 superscript 𝑒 imaginary-unit 𝜈 𝜋 2 Mathieu-me 𝜈 𝑧 1 2 𝜋 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,-q\right)=e^{\mathrm{i}% \nu\pi/2}\mathrm{me}_{\nu}\left(z-\tfrac{1}{2}\pi,q\right)}}
\Mathieume{\nu}@{z}{-q} = e^{\iunit\nu\pi/2}\Mathieume{\nu}@{z-\tfrac{1}{2}\pi}{q}		 

Error

Sqrt[2]*MathieuC[\[Nu], - q, z] == Exp[I*\[Nu]*Pi/2]*Sqrt[2]*MathieuC[\[Nu], q, z -Divide[1,2]*Pi]
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[1.2866300784936375, -3.291600297925931]
Test Values: {Rule[q, 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[-1.4943636299546066, 1.4617312701790142]
Test Values: {Rule[q, 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
28.12.E10 me ν ⁡ ( z , q ) ¯ = me ν ¯ ⁡ ( - z ¯ , q ¯ ) Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\overline{\mathrm{me}_{\nu}\left(z,q\right)}=% \mathrm{me}_{\overline{\nu}}\left(-\overline{z},\overline{q}\right)}}
\conj{\Mathieume{\nu}@{z}{q}} = \Mathieume{\conj{\nu}}@{-\conj{z}}{\conj{q}}		 

Error

Conjugate[Sqrt[2]*MathieuC[\[Nu], q, z]] == Sqrt[2]*MathieuC[Conjugate[\[Nu]], Conjugate[q], - Conjugate[z]]
Missing Macro Error Failure -
Failed [27 / 300]
Result: Complex[-1.449796041425081, -1.3521841059420128]
Test Values: {Rule[q, 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[-0.040892871573185774, -2.224553529597971]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
28.12#Ex1 me n ⁡ ( z , q ) = 2 ⁢ ce n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 2 Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{n}\left(z,q\right)=\sqrt{2}\mathrm{ce% }_{n}\left(z,q\right)}}
\Mathieume{n}@{z}{q} = \sqrt{2}\Mathieuce{n}@{z}{q}		 

Error

Sqrt[2]*MathieuC[n, q, z] == Sqrt[2]*MathieuC[n, q, z]
Missing Macro Error Successful - Successful [Tested: 70]
28.12#Ex2 me - n ⁡ ( z , q ) = - 2 ⁢ i ⁢ se n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 2 imaginary-unit Mathieu-se 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{-n}\left(z,q\right)=-\sqrt{2}\mathrm{% i}\mathrm{se}_{n}\left(z,q\right)}}
\Mathieume{-n}@{z}{q} = -\sqrt{2}\iunit\Mathieuse{n}@{z}{q}		 

Error

Sqrt[2]*MathieuC[- n, q, z] == -Sqrt[2]*I*MathieuS[n, q, z]
Missing Macro Error Failure -
Failed [210 / 210]
Result: Complex[-2.6058193733626913, 1.2555909202055446]
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[2.2564301512415783, 3.3896606696156866]
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
28.12.E12 ce ν ⁡ ( z , q ) = 1 2 ⁢ ( me ν ⁡ ( z , q ) + me ν ⁡ ( - z , q ) ) Mathieu-ce 𝜈 𝑧 𝑞 1 2 Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\tfrac{1}{2}% \left(\mathrm{me}_{\nu}\left(z,q\right)+\mathrm{me}_{\nu}\left(-z,q\right)% \right)}}
\Mathieuce{\nu}@{z}{q} = \tfrac{1}{2}\left(\Mathieume{\nu}@{z}{q}+\Mathieume{\nu}@{-z}{q}\right)		 

Error

MathieuC[\[Nu], q, z] == Divide[1,2]*(Sqrt[2]*MathieuC[\[Nu], q, z]+ Sqrt[2]*MathieuC[\[Nu], q, - z])
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-0.533819042119813, -0.14668719931348273]
Test Values: {Rule[q, 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[-0.3603013806161438, -0.6554927908359449]
Test Values: {Rule[q, 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
28.12.E13 se ν ⁡ ( z , q ) = - 1 2 ⁢ i ⁢ ( me ν ⁡ ( z , q ) - me ν ⁡ ( - z , q ) ) Mathieu-se 𝜈 𝑧 𝑞 1 2 imaginary-unit Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\tfrac{1}{2}% \mathrm{i}\left(\mathrm{me}_{\nu}\left(z,q\right)-\mathrm{me}_{\nu}\left(-z,q% \right)\right)}}
\Mathieuse{\nu}@{z}{q} = -\tfrac{1}{2}\iunit\left(\Mathieume{\nu}@{z}{q}-\Mathieume{\nu}@{-z}{q}\right)		 

Error

MathieuS[\[Nu], q, z] == -Divide[1,2]*I*(Sqrt[2]*MathieuC[\[Nu], q, z]- Sqrt[2]*MathieuC[\[Nu], q, - z])
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-0.5117296530175564, 1.1125419914222279]
Test Values: {Rule[q, 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[1.502309230543963, -0.7610291346347915]
Test Values: {Rule[q, 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
28.12.E14 ce ν ⁡ ( z , q ) = ce ν ⁡ ( - z , q ) Mathieu-ce 𝜈 𝑧 𝑞 Mathieu-ce 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\mathrm{ce}_{\nu% }\left(-z,q\right)}}
\Mathieuce{\nu}@{z}{q} = \Mathieuce{\nu}@{-z}{q}		 

MathieuCE(nu, q, z) = MathieuCE(nu, q, - z)

MathieuC[\[Nu], q, z] == MathieuC[\[Nu], q, - z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 300]
28.12.E14 ce ν ⁡ ( - z , q ) = ce - ν ⁡ ( z , q ) Mathieu-ce 𝜈 𝑧 𝑞 Mathieu-ce 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(-z,q\right)=\mathrm{ce}_{-% \nu}\left(z,q\right)}}
\Mathieuce{\nu}@{-z}{q} = \Mathieuce{-\nu}@{z}{q}		 

MathieuCE(nu, q, - z) = MathieuCE(- nu, q, z)

MathieuC[\[Nu], q, - z] == MathieuC[- \[Nu], q, z]
Failure Failure Error
Failed [300 / 300]
Result: Complex[2.1677458433372196, -0.552801794545088]
Test Values: {Rule[q, 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[0.1793065541346438, 1.5815013382691518]
Test Values: {Rule[q, 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
28.12.E15 se ν ⁡ ( z , q ) = - se ν ⁡ ( - z , q ) Mathieu-se 𝜈 𝑧 𝑞 Mathieu-se 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\mathrm{se}_{% \nu}\left(-z,q\right)}}
\Mathieuse{\nu}@{z}{q} = -\Mathieuse{\nu}@{-z}{q}		 

MathieuSE(nu, q, z) = - MathieuSE(nu, q, - z)

MathieuS[\[Nu], q, z] == - MathieuS[\[Nu], q, - z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 300]
28.12.E15 - se ν ⁡ ( - z , q ) = - se - ν ⁡ ( z , q ) Mathieu-se 𝜈 𝑧 𝑞 Mathieu-se 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle-\mathrm{se}_{\nu}\left(-z,q\right)=-\mathrm{se}_{% -\nu}\left(z,q\right)}}
-\Mathieuse{\nu}@{-z}{q} = -\Mathieuse{-\nu}@{z}{q}		 

- MathieuSE(nu, q, - z) = - MathieuSE(- nu, q, z)

- MathieuS[\[Nu], q, - z] == - MathieuS[- \[Nu], q, z]
Failure Failure Error
Failed [300 / 300]
Result: Complex[0.10223720739540931, 2.122915753327721]
Test Values: {Rule[q, 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[1.1209568079160426, 0.4323584529351461]
Test Values: {Rule[q, 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
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