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

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
28.29.E2	${\displaystyle{\displaystyle Q(z+\pi)=Q(z)}}$ Q(z+\pi) = Q(z)		Q(z + Pi) = Q(z)	Q[z + Pi] == Q[z]	Skipped - no semantic math	Skipped - no semantic math	-	-
28.29.E3	${\displaystyle{\displaystyle\int_{0}^{\pi}Q(z)\mathrm{d}z=0}}$ \int_{0}^{\pi}Q(z)\diff{z} = 0		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	${\displaystyle{\displaystyle-1<\Re\nu}}$ -1 < \realpart@@{\nu}		- 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	${\displaystyle{\displaystyle\Re\nu\leq 1}}$ \realpart@@{\nu} \leq 1		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	${\displaystyle{\displaystyle w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z)}}$ 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
28.29.E11	${\displaystyle{\displaystyle w(z+\pi)=(-1)^{\nu}w(z)+cP(z)}}$ w(z+\pi) = (-1)^{\nu}w(z)+cP(z)		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	${\displaystyle{\displaystyle w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z)}}$ 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
28.29.E18	${\displaystyle{\displaystyle\lambda_{0}<\mu_{1}}}$ \lambda_{0} < \mu_{1}		lambda[0](<)*mu[1]	Subscript[\[Lambda], 0][<]*Subscript[\[Mu], 1]	Skipped - no semantic math	Skipped - no semantic math	-	-