Painlevé Transcendents - 32.10 Special Function Solutions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.10.E2 α = n + 1 2 𝛼 𝑛 1 2 {\displaystyle{\displaystyle\alpha=n+\tfrac{1}{2}}}
\alpha = n+\tfrac{1}{2}		 

alpha = n +(1)/(2)
 
\[Alpha] == n +Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E5 ϕ ⁢ ( z ) = C 1 ⁢ Ai ⁡ ( - 2 - 1 / 3 ⁢ z ) + C 2 ⁢ Bi ⁡ ( - 2 - 1 / 3 ⁢ z ) italic-ϕ 𝑧 subscript 𝐶 1 Airy-Ai superscript 2 1 3 𝑧 subscript 𝐶 2 Airy-Bi superscript 2 1 3 𝑧 {\displaystyle{\displaystyle\phi(z)=C_{1}\mathrm{Ai}\left(-2^{-1/3}z\right)+C_% {2}\mathrm{Bi}\left(-2^{-1/3}z\right)}}
\phi(z) = C_{1}\AiryAi@{-2^{-1/3}z}+C_{2}\AiryBi@{-2^{-1/3}z}		 

phi(z) = C[1]*AiryAi(- (2)^(- 1/3)* z)+ C[2]*AiryBi(- (2)^(- 1/3)* z)

\[Phi][z] == Subscript[C, 1]*AiryAi[- (2)^(- 1/3)* z]+ Subscript[C, 2]*AiryBi[- (2)^(- 1/3)* z]
Failure Failure
Failed [300 / 300]
Result: -.2986692739+.5787509238*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = 1/2*3^(1/2)+1/2*I}


Result: .3018910357e-1+.1740730853*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.29866927421000106, 0.5787509234724151]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.03018910341830547, 0.174073084997731]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.10.E6 w ⁢ ( z ; 3 2 ) = Φ - 1 2 ⁢ Φ 2 + z 𝑤 𝑧 3 2 Φ 1 2 superscript Φ 2 𝑧 {\displaystyle{\displaystyle w(z;\tfrac{3}{2})=\Phi-\dfrac{1}{2\Phi^{2}+z}}}
w(z;\tfrac{3}{2}) = \Phi-\dfrac{1}{2\Phi^{2}+z}		 

w(z ;(3)/(2)) = Phi -(1)/(2*(Phi)^(2)+ z)
 
w[z ;Divide[3,2]] == \[CapitalPhi]-Divide[1,2*\[CapitalPhi]^(2)+ z]
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E7 w ⁢ ( z ; 5 2 ) = 1 2 ⁢ Φ 2 + z + 2 ⁢ z ⁢ Φ 2 + Φ + z 2 4 ⁢ Φ 3 + 2 ⁢ z ⁢ Φ - 1 𝑤 𝑧 5 2 1 2 superscript Φ 2 𝑧 2 𝑧 superscript Φ 2 Φ superscript 𝑧 2 4 superscript Φ 3 2 𝑧 Φ 1 {\displaystyle{\displaystyle w(z;\tfrac{5}{2})=\dfrac{1}{2\Phi^{2}+z}+\dfrac{2% z\Phi^{2}+\Phi+z^{2}}{4\Phi^{3}+2z\Phi-1}}}
w(z;\tfrac{5}{2}) = \dfrac{1}{2\Phi^{2}+z}+\dfrac{2z\Phi^{2}+\Phi+z^{2}}{4\Phi^{3}+2z\Phi-1}		 

w(z ;(5)/(2)) = (1)/(2*(Phi)^(2)+ z)+(2*z*(Phi)^(2)+ Phi + (z)^(2))/(4*(Phi)^(3)+ 2*z*Phi - 1)
 
w[z ;Divide[5,2]] == Divide[1,2*\[CapitalPhi]^(2)+ z]+Divide[2*z*\[CapitalPhi]^(2)+ \[CapitalPhi]+ (z)^(2),4*\[CapitalPhi]^(3)+ 2*z*\[CapitalPhi]- 1]
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E8 w ⁢ ( z ; n + 1 2 ) = d d z ⁡ ( ln ⁡ ( τ n ⁢ ( z ) τ n + 1 ⁢ ( z ) ) ) 𝑤 𝑧 𝑛 1 2 derivative 𝑧 subscript 𝜏 𝑛 𝑧 subscript 𝜏 𝑛 1 𝑧 {\displaystyle{\displaystyle w(z;n+\tfrac{1}{2})=\frac{\mathrm{d}}{\mathrm{d}z% }\left(\ln\left(\frac{\tau_{n}(z)}{\tau_{n+1}(z)}\right)\right)}}
w(z;n+\tfrac{1}{2}) = \deriv{}{z}\left(\ln@{\frac{\tau_{n}(z)}{\tau_{n+1}(z)}}\right)		 

w(z ; n +(1)/(2)) = diff(ln((tau[n](z))/(tau[n + 1](z))), z)

w[z ; n +Divide[1,2]] == D[Log[Divide[Subscript[\[Tau], n][z],Subscript[\[Tau], n + 1][z]]], z]
Translation Error Translation Error - -
32.10.E10 w ⁢ ( z ; - n - 1 2 ) = - w ⁢ ( z ; n + 1 2 ) 𝑤 𝑧 𝑛 1 2 𝑤 𝑧 𝑛 1 2 {\displaystyle{\displaystyle w(z;-n-\tfrac{1}{2})=-w(z;n+\tfrac{1}{2})}}
w(z;-n-\tfrac{1}{2}) = -w(z;n+\tfrac{1}{2})		 

w(z ; - n -(1)/(2)) = - w(z ; n +(1)/(2))
 
w[z ; - n -Divide[1,2]] == - w[z ; n +Divide[1,2]]
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E11 ε 1 ⁢ α + ε 2 ⁢ β = 4 ⁢ n + 2 subscript 𝜀 1 𝛼 subscript 𝜀 2 𝛽 4 𝑛 2 {\displaystyle{\displaystyle\varepsilon_{1}\alpha+\varepsilon_{2}\beta=4n+2}}
\varepsilon_{1}\alpha+\varepsilon_{2}\beta = 4n+2		 

varepsilon[1]*alpha + varepsilon[2]*beta = 4*n + 2
 
Subscript[\[CurlyEpsilon], 1]*\[Alpha]+ Subscript[\[CurlyEpsilon], 2]*\[Beta] == 4*n + 2
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E14 ϕ ⁢ ( z ) = z ν ⁢ ( C 1 ⁢ J ν ⁡ ( ζ ) + C 2 ⁢ Y ν ⁡ ( ζ ) ) italic-ϕ 𝑧 superscript 𝑧 𝜈 subscript 𝐶 1 Bessel-J 𝜈 𝜁 subscript 𝐶 2 Bessel-Y-Weber 𝜈 𝜁 {\displaystyle{\displaystyle\phi(z)=z^{\nu}\left(C_{1}J_{\nu}\left(\zeta\right% )+C_{2}Y_{\nu}\left(\zeta\right)\right)}}
\phi(z) = z^{\nu}\left(C_{1}\BesselJ{\nu}@{\zeta}+C_{2}\BesselY{\nu}@{\zeta}\right)		 ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( - ν ) + k + 1 ) > 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}} 
phi(z) = (z)^(nu)*(C[1]*BesselJ(nu, zeta)+ C[2]*BesselY(nu, zeta))

\[Phi][z] == (z)^\[Nu]*(Subscript[C, 1]*BesselJ[\[Nu], \[Zeta]]+ Subscript[C, 2]*BesselY[\[Nu], \[Zeta]])
Failure Failure
Failed [300 / 300]
Result: .6857713611+1.049278090*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = 1/2*3^(1/2)+1/2*I}


Result: -.1639325500+1.038275666*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6857713606630202, 1.0492780901981935]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.16393255022963316, 1.0382756660889538]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.10.E15 β = - 2 ⁢ ( 2 ⁢ n + 1 + ε ⁢ α ) 2 𝛽 2 superscript 2 𝑛 1 𝜀 𝛼 2 {\displaystyle{\displaystyle\beta=-2(2n+1+\varepsilon\alpha)^{2}}}
\beta = -2(2n+1+\varepsilon\alpha)^{2}		 

beta = - 2*(2*n + 1 + varepsilon*alpha)^(2)
 
\[Beta] == - 2*(2*n + 1 + \[CurlyEpsilon]*\[Alpha])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E16 β = - 2 ⁢ n 2 𝛽 2 superscript 𝑛 2 {\displaystyle{\displaystyle\beta=-2n^{2}}}
\beta = -2n^{2}		 

beta = - 2*(n)^(2)
 
\[Beta] == - 2*(n)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E19 ϕ ⁢ ( z ) = ( C 1 ⁢ U ⁡ ( a , 2 ⁢ z ) + C 2 ⁢ V ⁡ ( a , 2 ⁢ z ) ) ⁢ exp ⁡ ( 1 2 ⁢ ε ⁢ z 2 ) italic-ϕ 𝑧 subscript 𝐶 1 parabolic-U 𝑎 2 𝑧 subscript 𝐶 2 parabolic-V 𝑎 2 𝑧 1 2 𝜀 superscript 𝑧 2 {\displaystyle{\displaystyle\phi(z)=\left(C_{1}U\left(a,\sqrt{2}z\right)+C_{2}% V\left(a,\sqrt{2}z\right)\right)\exp\left(\tfrac{1}{2}\varepsilon z^{2}\right)}}
\phi(z) = \left(C_{1}\paraU@{a}{\sqrt{2}z}+C_{2}\paraV@{a}{\sqrt{2}z}\right)\exp@{\tfrac{1}{2}\varepsilon z^{2}}		 

phi(z) = (C[1]*CylinderU(a, sqrt(2)*z)+ C[2]*CylinderV(a, sqrt(2)*z))*exp((1)/(2)*varepsilon*(z)^(2))

\[Phi][z] == (Subscript[C, 1]*ParabolicCylinderD[- 1/2 -(a), Sqrt[2]*z]+ Subscript[C, 2]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, Sqrt[2]*z] + ParabolicCylinderD[-(a) - 1/2, -(Sqrt[2]*z)]))*Exp[Divide[1,2]*\[CurlyEpsilon]*(z)^(2)]
Failure Failure
Failed [300 / 300]
Result: .6213533818-.8057984780*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = 1/2*3^(1/2)+1/2*I, varepsilon = 1}


Result: 1.542195596-1.017130546*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = 1/2*3^(1/2)+1/2*I, varepsilon = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-1.165517214154348, -0.5387865015105858], Plus[Complex[0.9001043151387932, 0.6347232005321619], Times[Complex[-6.562724044143109*^-17, -2.768827103772538*^-17], GAMMA[-1.0]]]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-1.0681394822800956, -1.2559298845291706], Plus[Complex[0.9001043151387932, 0.6347232005321619], Times[Complex[-6.562724044143109*^-17, -2.768827103772538*^-17], GAMMA[-1.0]]]]]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ε, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.10.E20 w ⁢ ( z ; - m , - 2 ⁢ ( m - 1 ) 2 ) = - H m - 1 ′ ⁡ ( z ) H m - 1 ⁡ ( z ) 𝑤 𝑧 𝑚 2 superscript 𝑚 1 2 diffop Hermite-polynomial-H 𝑚 1 1 𝑧 Hermite-polynomial-H 𝑚 1 𝑧 {\displaystyle{\displaystyle w(z;-m,-2(m-1)^{2})=-\frac{H_{m-1}'\left(z\right)% }{H_{m-1}\left(z\right)}}}
w(z;-m,-2(m-1)^{2}) = -\frac{\HermitepolyH{m-1}'@{z}}{\HermitepolyH{m-1}@{z}}		 

w(z ; - m , - 2*(m - 1)^(2)) = -(diff( HermiteH(m - 1, z), z$(1) ))/(HermiteH(m - 1, z))

w[z ; - m , - 2*(m - 1)^(2)] == -Divide[D[HermiteH[m - 1, z], {z, 1}],HermiteH[m - 1, z]]
Translation Error Translation Error - -
32.10.E21 w ⁢ ( z ; - m , - 2 ⁢ ( m + 1 ) 2 ) = - 2 ⁢ z + H m ′ ⁡ ( z ) H m ⁡ ( z ) 𝑤 𝑧 𝑚 2 superscript 𝑚 1 2 2 𝑧 diffop Hermite-polynomial-H 𝑚 1 𝑧 Hermite-polynomial-H 𝑚 𝑧 {\displaystyle{\displaystyle w(z;-m,-2(m+1)^{2})=-2z+\frac{H_{m}'\left(z\right% )}{H_{m}\left(z\right)}}}
w(z;-m,-2(m+1)^{2}) = -2z+\frac{\HermitepolyH{m}'@{z}}{\HermitepolyH{m}@{z}}		 

w(z ; - m , - 2*(m + 1)^(2)) = - 2*z +(diff( HermiteH(m, z), z$(1) ))/(HermiteH(m, z))

w[z ; - m , - 2*(m + 1)^(2)] == - 2*z +Divide[D[HermiteH[m, z], {z, 1}],HermiteH[m, z]]
Translation Error Translation Error - -
32.10.E23 a + b + ε 3 ⁢ γ = 2 ⁢ n + 1 𝑎 𝑏 subscript 𝜀 3 𝛾 2 𝑛 1 {\displaystyle{\displaystyle a+b+\varepsilon_{3}\gamma=2n+1}}
a+b+\varepsilon_{3}\gamma = 2n+1		 

a + b + varepsilon[3]*gamma = 2*n + 1
 
a + b + Subscript[\[CurlyEpsilon], 3]*\[Gamma] == 2*n + 1
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E24 ( a - n ) ⁢ ( b - n ) = 0 𝑎 𝑛 𝑏 𝑛 0 {\displaystyle{\displaystyle(a-n)(b-n)=0}}
(a-n)(b-n) = 0		 

(a - n)*(b - n) = 0
 
(a - n)*(b - n) == 0
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E27 ϕ ⁢ ( z ) = C 1 ⁢ M κ , μ ⁡ ( ζ ) + C 2 ⁢ W κ , μ ⁡ ( ζ ) ζ ( a - b + 1 ) / 2 ⁢ exp ⁡ ( 1 2 ⁢ ζ ) italic-ϕ 𝑧 subscript 𝐶 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝜁 subscript 𝐶 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝜁 superscript 𝜁 𝑎 𝑏 1 2 1 2 𝜁 {\displaystyle{\displaystyle\phi(z)=\frac{C_{1}M_{\kappa,\mu}\left(\zeta\right% )+C_{2}W_{\kappa,\mu}\left(\zeta\right)}{\zeta^{(a-b+1)/2}}\exp\left(\tfrac{1}% {2}\zeta\right)}}
\phi(z) = \frac{C_{1}\WhittakerconfhyperM{\kappa}{\mu}@{\zeta}+C_{2}\WhittakerconfhyperW{\kappa}{\mu}@{\zeta}}{\zeta^{(a-b+1)/2}}\exp@{\tfrac{1}{2}\zeta}		 

phi(z) = (C[1]*WhittakerM(kappa, mu, zeta)+ C[2]*WhittakerW(kappa, mu, zeta))/((zeta)^((a - b + 1)/2))*exp((1)/(2)*zeta)

\[Phi][z] == Divide[Subscript[C, 1]*WhittakerM[\[Kappa], \[Mu], \[Zeta]]+ Subscript[C, 2]*WhittakerW[\[Kappa], \[Mu], \[Zeta]],\[Zeta]^((a - b + 1)/2)]*Exp[Divide[1,2]*\[Zeta]]
Failure Failure Manual Skip! Skipped - Because timed out
32.10.E28 a + b + c + d = 2 ⁢ n + 1 𝑎 𝑏 𝑐 𝑑 2 𝑛 1 {\displaystyle{\displaystyle a+b+c+d=2n+1}}
a+b+c+d = 2n+1		 

a + b + c + d = 2*n + 1
 
a + b + c + d == 2*n + 1
 Skipped - no semantic math Skipped - no semantic math - -
32.10#Ex2 ζ = 1 1 - z 𝜁 1 1 𝑧 {\displaystyle{\displaystyle\zeta=\frac{1}{1-z}}}
\zeta = \frac{1}{1-z}		 

zeta = (1)/(1 - z)
 
\[Zeta] == Divide[1,1 - z]
 Skipped - no semantic math Skipped - no semantic math - -
32.10.E31 ϕ ⁢ ( ζ ) = C 1 ⁢ F ⁡ ( b , - a ; b + c ; ζ ) + C 2 ⁢ ζ - b + 1 - c ⁢ F ⁡ ( - a - b - c + 1 , - c + 1 ; 2 - b - c ; ζ ) italic-ϕ 𝜁 subscript 𝐶 1 Gauss-hypergeometric-F 𝑏 𝑎 𝑏 𝑐 𝜁 subscript 𝐶 2 superscript 𝜁 𝑏 1 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑐 1 2 𝑏 𝑐 𝜁 {\displaystyle{\displaystyle\phi(\zeta)=C_{1}F\left(b,-a;b+c;\zeta\right)+C_{2% }\zeta^{-b+1-c}\*F\left(-a-b-c+1,-c+1;2-b-c;\zeta\right)}}
\phi(\zeta) = C_{1}\hyperF@{b}{-a}{b+c}{\zeta}+C_{2}\zeta^{-b+1-c}\*\hyperF@{-a-b-c+1}{-c+1}{2-b-c}{\zeta}		 

phi(zeta) = C[1]*hypergeom([b, - a], [b + c], zeta)+ C[2]*(zeta)^(- b + 1 - c)* hypergeom([- a - b - c + 1, - c + 1], [2 - b - c], zeta)

\[Phi][\[Zeta]] == Subscript[C, 1]*Hypergeometric2F1[b, - a, b + c, \[Zeta]]+ Subscript[C, 2]*\[Zeta]^(- b + 1 - c)* Hypergeometric2F1[- a - b - c + 1, - c + 1, 2 - b - c, \[Zeta]]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, phi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = 1/2*3^(1/2)+1/2*I}


Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, phi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, C[1] = 1/2*3^(1/2)+1/2*I, C[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
32.10.E32 u = ∫ 0 Λ d t t ⁢ ( t - 1 ) ⁢ ( t - z ) 𝑢 superscript subscript 0 Λ 𝑡 𝑡 𝑡 1 𝑡 𝑧 {\displaystyle{\displaystyle u=\int_{0}^{\Lambda}\frac{\mathrm{d}t}{\sqrt{t(t-% 1)(t-z)}}}}
u = \int_{0}^{\Lambda}\frac{\diff{t}}{\sqrt{t(t-1)(t-z)}}		 

u = int((1)/(sqrt(t*(t - 1)*(t - z))), t = 0..Lambda)

u == Integrate[Divide[1,Sqrt[t*(t - 1)*(t - z)]], {t, 0, \[CapitalLambda]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
32.10.E33 z ⁢ ( 1 - z ) ⁢ d 2 ϕ d z 2 + ( 1 - 2 ⁢ z ) ⁢ d ϕ d z - 1 4 ⁢ ϕ = 0 𝑧 1 𝑧 derivative italic-ϕ 𝑧 2 1 2 𝑧 derivative italic-ϕ 𝑧 1 4 italic-ϕ 0 {\displaystyle{\displaystyle z(1-z)\frac{{\mathrm{d}}^{2}\phi}{{\mathrm{d}z}^{% 2}}+(1-2z)\frac{\mathrm{d}\phi}{\mathrm{d}z}-\tfrac{1}{4}\phi=0}}
z(1-z)\deriv[2]{\phi}{z}+(1-2z)\deriv{\phi}{z}-\tfrac{1}{4}\phi = 0		 

z*(1 - z)*diff(phi, [z$(2)])+(1 - 2*z)*diff(phi, z)-(1)/(4)*phi = 0

z*(1 - z)*D[\[Phi], {z, 2}]+(1 - 2*z)*D[\[Phi], z]-Divide[1,4]*\[Phi] == 0
Failure Failure
Failed [70 / 70]
Result: -.2165063510-.1250000000*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.2165063510-.1250000000*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[-0.21650635094610968, -0.12499999999999999]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.12499999999999994, -0.21650635094610968]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.10.E34 w ⁢ ( z ; 0 , 0 , 0 , 1 2 ) = Λ ⁢ ( C 1 ⁢ ϕ 1 + C 2 ⁢ ϕ 2 , z ) 𝑤 𝑧 0 0 0 1 2 Λ subscript 𝐶 1 subscript italic-ϕ 1 subscript 𝐶 2 subscript italic-ϕ 2 𝑧 {\displaystyle{\displaystyle w(z;0,0,0,\tfrac{1}{2})=\Lambda(C_{1}\phi_{1}+C_{% 2}\phi_{2},z)}}
w(z;0,0,0,\tfrac{1}{2}) = \Lambda(C_{1}\phi_{1}+C_{2}\phi_{2},z)		 

w(z ; 0 , 0 , 0 ,(1)/(2)) = Lambda(C[1]*phi[1]+ C[2]*phi[2], z)
 
w[z ; 0 , 0 , 0 ,Divide[1,2]] == \[CapitalLambda][Subscript[C, 1]*Subscript[\[Phi], 1]+ Subscript[C, 2]*Subscript[\[Phi], 2], z]
 Skipped - no semantic math Skipped - no semantic math - -
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