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 w ⁢ ( z + π ) = e π ⁢ i ⁢ ν ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 superscript 𝑒 𝜋 imaginary-unit 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z)}}
w(z+\pi) = e^{\pi\iunit\nu}w(z)		 

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 w ⁢ ( z + π ) + w ⁢ ( z - π ) = 2 ⁢ cos ⁡ ( π ⁢ ν ) ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 𝑤 𝑧 𝜋 2 𝜋 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z)}}
w(z+\pi)+w(z-\pi) = 2\cos@{\pi\nu}w(z)		 

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 w ⁢ ( z ) = ∑ n = - ∞ ∞ c 2 ⁢ n ⁢ e i ⁢ ( ν + 2 ⁢ n ) ⁢ z 𝑤 𝑧 superscript subscript 𝑛 subscript 𝑐 2 𝑛 superscript 𝑒 imaginary-unit 𝜈 2 𝑛 𝑧 {\displaystyle{\displaystyle w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}% (\nu+2n)z}}}
w(z) = \sum_{n=-\infty}^{\infty}c_{2n}e^{\iunit(\nu+2n)z}		 

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 q ⁢ c 2 ⁢ n + 2 - ( a - ( ν + 2 ⁢ n ) 2 ) ⁢ c 2 ⁢ n + q ⁢ c 2 ⁢ n - 2 = 0 , 𝑞 subscript 𝑐 2 𝑛 2 𝑎 superscript 𝜈 2 𝑛 2 subscript 𝑐 2 𝑛 𝑞 subscript 𝑐 2 𝑛 2 0 {\displaystyle{\displaystyle qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2% n-2}=0,}}
qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2} = 0,		 

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 lim n → + ∞ ⁡ | c 2 ⁢ n | 1 / | n | = 0 subscript → 𝑛 superscript subscript 𝑐 2 𝑛 1 𝑛 0 {\displaystyle{\displaystyle\lim_{n\to+\infty}|c_{2n}|^{1/|n|}=0}}
\lim_{n\to+\infty}|c_{2n}|^{1/|n|} = 0		 

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 a n ⁡ ( 0 ) = n 2 Mathieu-eigenvalue-a 𝑛 0 superscript 𝑛 2 {\displaystyle{\displaystyle a_{n}\left(0\right)=n^{2}}}
\Mathieueigvala{n}@{0} = n^{2}		 

MathieuA(n, 0) = (n)^(2)

MathieuCharacteristicA[n, 0] == (n)^(2)
Successful Successful - Successful [Tested: 1]
28.2.E24 b n ⁡ ( 0 ) = n 2 Mathieu-eigenvalue-b 𝑛 0 superscript 𝑛 2 {\displaystyle{\displaystyle b_{n}\left(0\right)=n^{2}}}
\Mathieueigvalb{n}@{0} = n^{2}		 

MathieuB(n, 0) = (n)^(2)

MathieuCharacteristicB[n, 0] == (n)^(2)
Successful Successful - Successful [Tested: 1]
28.2.E26 a 2 ⁢ n ⁡ ( - q ) = a 2 ⁢ n ⁡ ( q ) Mathieu-eigenvalue-a 2 𝑛 𝑞 Mathieu-eigenvalue-a 2 𝑛 𝑞 {\displaystyle{\displaystyle a_{2n}\left(-q\right)=a_{2n}\left(q\right)}}
\Mathieueigvala{2n}@{-q} = \Mathieueigvala{2n}@{q}		 

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 a 2 ⁢ n + 1 ⁡ ( - q ) = b 2 ⁢ n + 1 ⁡ ( q ) Mathieu-eigenvalue-a 2 𝑛 1 𝑞 Mathieu-eigenvalue-b 2 𝑛 1 𝑞 {\displaystyle{\displaystyle a_{2n+1}\left(-q\right)=b_{2n+1}\left(q\right)}}
\Mathieueigvala{2n+1}@{-q} = \Mathieueigvalb{2n+1}@{q}		 

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 b 2 ⁢ n + 2 ⁡ ( - q ) = b 2 ⁢ n + 2 ⁡ ( q ) Mathieu-eigenvalue-b 2 𝑛 2 𝑞 Mathieu-eigenvalue-b 2 𝑛 2 𝑞 {\displaystyle{\displaystyle b_{2n+2}\left(-q\right)=b_{2n+2}\left(q\right)}}
\Mathieueigvalb{2n+2}@{-q} = \Mathieueigvalb{2n+2}@{q}		 

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 ce 0 ⁡ ( z , 0 ) = 1 / 2 Mathieu-ce 0 𝑧 0 1 2 {\displaystyle{\displaystyle\mathrm{ce}_{0}\left(z,0\right)=1/\sqrt{2}}}
\Mathieuce{0}@{z}{0} = 1/\sqrt{2}		 

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 ce n ⁡ ( z , 0 ) = cos ⁡ ( n ⁢ z ) Mathieu-ce 𝑛 𝑧 0 𝑛 𝑧 {\displaystyle{\displaystyle\mathrm{ce}_{n}\left(z,0\right)=\cos\left(nz\right% )}}
\Mathieuce{n}@{z}{0} = \cos@{nz}		 

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 se n ⁡ ( z , 0 ) = sin ⁡ ( n ⁢ z ) Mathieu-se 𝑛 𝑧 0 𝑛 𝑧 {\displaystyle{\displaystyle\mathrm{se}_{n}\left(z,0\right)=\sin\left(nz\right% )}}
\Mathieuse{n}@{z}{0} = \sin@{nz}		 

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 ∫ 0 2 ⁢ π ( ce n ⁡ ( x , q ) ) 2 ⁢ d x = π superscript subscript 0 2 𝜋 superscript Mathieu-ce 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{2\pi}\left(\mathrm{ce}_{n}\left(x,q% \right)\right)^{2}\mathrm{d}x=\pi}}
\int_{0}^{2\pi}\left(\Mathieuce{n}@{x}{q}\right)^{2}\diff{x} = \pi		 

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 ∫ 0 2 ⁢ π ( se n ⁡ ( x , q ) ) 2 ⁢ d x = π superscript subscript 0 2 𝜋 superscript Mathieu-se 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{2\pi}\left(\mathrm{se}_{n}\left(x,q% \right)\right)^{2}\mathrm{d}x=\pi}}
\int_{0}^{2\pi}\left(\Mathieuse{n}@{x}{q}\right)^{2}\diff{x} = \pi		 

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 ∫ 0 2 ⁢ π ce m ⁡ ( x , q ) ⁢ ce n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-ce 𝑚 𝑥 𝑞 Mathieu-ce 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{ce}_{m}\left(x,q\right)% \mathrm{ce}_{n}\left(x,q\right)\mathrm{d}x=0}}
\int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuce{n}@{x}{q}\diff{x} = 0		 n ≠ m 𝑛 𝑚 {\displaystyle{\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 ∫ 0 2 ⁢ π se m ⁡ ( x , q ) ⁢ se n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-se 𝑚 𝑥 𝑞 Mathieu-se 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{se}_{m}\left(x,q\right)% \mathrm{se}_{n}\left(x,q\right)\mathrm{d}x=0}}
\int_{0}^{2\pi}\Mathieuse{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0		 n ≠ m 𝑛 𝑚 {\displaystyle{\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 ∫ 0 2 ⁢ π ce m ⁡ ( x , q ) ⁢ se n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-ce 𝑚 𝑥 𝑞 Mathieu-se 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{ce}_{m}\left(x,q\right)% \mathrm{se}_{n}\left(x,q\right)\mathrm{d}x=0}}
\int_{0}^{2\pi}\Mathieuce{m}@{x}{q}\Mathieuse{n}@{x}{q}\diff{x} = 0		 

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 ce 2 ⁢ n ⁡ ( z , - q ) = ( - 1 ) n ⁢ ce 2 ⁢ n ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-ce 2 𝑛 𝑧 𝑞 superscript 1 𝑛 Mathieu-ce 2 𝑛 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{2n}\left(z,-q\right)=(-1)^{n}\mathrm{% ce}_{2n}\left(\tfrac{1}{2}\pi-z,q\right)}}
\Mathieuce{2n}@{z}{-q} = (-1)^{n}\Mathieuce{2n}@{\tfrac{1}{2}\pi-z}{q}		 

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 ce 2 ⁢ n + 1 ⁡ ( z , - q ) = ( - 1 ) n ⁢ se 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-ce 2 𝑛 1 𝑧 𝑞 superscript 1 𝑛 Mathieu-se 2 𝑛 1 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{2n+1}\left(z,-q\right)=(-1)^{n}% \mathrm{se}_{2n+1}\left(\tfrac{1}{2}\pi-z,q\right)}}
\Mathieuce{2n+1}@{z}{-q} = (-1)^{n}\Mathieuse{2n+1}@{\tfrac{1}{2}\pi-z}{q}		 

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 se 2 ⁢ n + 1 ⁡ ( z , - q ) = ( - 1 ) n ⁢ ce 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-se 2 𝑛 1 𝑧 𝑞 superscript 1 𝑛 Mathieu-ce 2 𝑛 1 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{2n+1}\left(z,-q\right)=(-1)^{n}% \mathrm{ce}_{2n+1}\left(\tfrac{1}{2}\pi-z,q\right)}}
\Mathieuse{2n+1}@{z}{-q} = (-1)^{n}\Mathieuce{2n+1}@{\tfrac{1}{2}\pi-z}{q}		 

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 se 2 ⁢ n + 2 ⁡ ( z , - q ) = ( - 1 ) n ⁢ se 2 ⁢ n + 2 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-se 2 𝑛 2 𝑧 𝑞 superscript 1 𝑛 Mathieu-se 2 𝑛 2 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{2n+2}\left(z,-q\right)=(-1)^{n}% \mathrm{se}_{2n+2}\left(\tfrac{1}{2}\pi-z,q\right)}}
\Mathieuse{2n+2}@{z}{-q} = (-1)^{n}\Mathieuse{2n+2}@{\tfrac{1}{2}\pi-z}{q}		 

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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