28.12

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
28.12.E4	${\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,0\right)=e^{\mathrm{i}\nu z}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\pi}\mathrm{me}_{\nu}\left(x,q\right)\mathrm{me}_{\nu}\left(-x,q\right)\mathrm{d}x=\pi}}$ \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	${\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z+\pi,q\right)=e^{\pi\mathrm{i}\nu}\mathrm{me}_{\nu}\left(z,q\right)}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\pi}\mathrm{me}_{\nu+2m}\left(x,q\right)\mathrm{me}_{\nu+2n}\left(-x,q\right)\mathrm{d}x=0,}}$ \int_{0}^{\pi}\Mathieume{\nu+2m}@{x}{q}\Mathieume{\nu+2n}@{-x}{q}\diff{x} = 0,	${\displaystyle{\displaystyle m\neq n}}$	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	${\displaystyle{\displaystyle\mathrm{me}_{-\nu}\left(z,q\right)=\mathrm{me}_{\nu}\left(-z,q\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,-q\right)=e^{\mathrm{i}\nu\pi/2}\mathrm{me}_{\nu}\left(z-\tfrac{1}{2}\pi,q\right)}}$ \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	${\displaystyle{\displaystyle\overline{\mathrm{me}_{\nu}\left(z,q\right)}=\mathrm{me}_{\overline{\nu}}\left(-\overline{z},\overline{q}\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{me}_{n}\left(z,q\right)=\sqrt{2}\mathrm{ce}_{n}\left(z,q\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{me}_{-n}\left(z,q\right)=-\sqrt{2}\mathrm{i}\mathrm{se}_{n}\left(z,q\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\tfrac{1}{2}\left(\mathrm{me}_{\nu}\left(z,q\right)+\mathrm{me}_{\nu}\left(-z,q\right)\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\tfrac{1}{2}\mathrm{i}\left(\mathrm{me}_{\nu}\left(z,q\right)-\mathrm{me}_{\nu}\left(-z,q\right)\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\mathrm{ce}_{\nu}\left(-z,q\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(-z,q\right)=\mathrm{ce}_{-\nu}\left(z,q\right)}}$ \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	${\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\mathrm{se}_{\nu}\left(-z,q\right)}}$ \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	${\displaystyle{\displaystyle-\mathrm{se}_{\nu}\left(-z,q\right)=-\mathrm{se}_{-\nu}\left(z,q\right)}}$ -\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