Painlevé Transcendents - 32.11 Asymptotic Approximations for Real Variables

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.11.E2 ϕ ⁢ ( x ) = ( 24 ) 1 / 4 ⁢ ( 4 5 ⁢ | x | 5 / 4 - 5 8 ⁢ d 2 ⁢ ln ⁡ | x | ) italic-ϕ 𝑥 superscript 24 1 4 4 5 superscript 𝑥 5 4 5 8 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=(24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-% \tfrac{5}{8}d^{2}\ln|x|\right)}}
\phi(x) = (24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-\tfrac{5}{8}d^{2}\ln@@{|x|}\right)		 

phi(x) = (24)^(1/4)*((4)/(5)*(abs(x))^(5/4)-(5)/(8)*(d)^(2)* ln(abs(x)))

\[Phi][x] == (24)^(1/4)*(Divide[4,5]*(Abs[x])^(5/4)-Divide[5,8]*(d)^(2)* Log[Abs[x]])
Failure Failure
Failed [300 / 300]
Result: -1.359899020+1.235754628*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.7909045934-.5804030210*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.3598990205302544, 1.2357546278215892]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.408937126206912, 1.7847927334982474]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E7 ϕ ⁢ ( x ) = 2 3 ⁢ | x | 3 / 2 - 3 4 ⁢ d 2 ⁢ ln ⁡ | x | italic-ϕ 𝑥 2 3 superscript 𝑥 3 2 3 4 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{2}{3}|x|^{3/2}-\tfrac{3}{4}d^{2}\ln% |x|}}
\phi(x) = \tfrac{2}{3}|x|^{3/2}-\tfrac{3}{4}d^{2}\ln@@{|x|}		 

phi(x) = (2)/(3)*(abs(x))^(3/2)-(3)/(4)*(d)^(2)* ln(abs(x))

\[Phi][x] == Divide[2,3]*(Abs[x])^(3/2)-Divide[3,4]*(d)^(2)* Log[Abs[x]]
Failure Failure
Failed [300 / 300]
Result: .2263426496+1.013357313*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.626197514e-1-.2002123003*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.22634264982563074, 1.0133573129774054]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.8226954558510269, 1.5623954186540636]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E8 d 2 = - π - 1 ⁢ ln ⁡ ( 1 - k 2 ) superscript 𝑑 2 superscript 𝜋 1 1 superscript 𝑘 2 {\displaystyle{\displaystyle d^{2}=-\pi^{-1}\ln\left(1-k^{2}\right)}}
d^{2} = -\pi^{-1}\ln@{1-k^{2}}		 

(d)^(2) = - (Pi)^(- 1)* ln(1 - (k)^(2))

(d)^(2) == - (Pi)^(- 1)* Log[1 - (k)^(2)]
Failure Failure
Failed [30 / 30]
Result: Float(-infinity)+.8660254040*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .8496991530+1.866025404*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: DirectedInfinity[-1]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1]}


Result: Complex[0.84969915256606, 1.8660254037844386]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2]}

... skip entries to safe data
32.11.E9 θ 0 = 3 2 ⁢ d 2 ⁢ ln ⁡ 2 + ph ⁡ Γ ⁡ ( 1 - 1 2 ⁢ i ⁢ d 2 ) + 1 4 ⁢ π ⁢ ( 1 - 2 ⁢ sign ⁡ ( k ) ) subscript 𝜃 0 3 2 superscript 𝑑 2 2 phase Euler-Gamma 1 1 2 𝑖 superscript 𝑑 2 1 4 𝜋 1 2 sign 𝑘 {\displaystyle{\displaystyle\theta_{0}=\tfrac{3}{2}d^{2}\ln 2+\operatorname{ph% }\Gamma\left(1-\tfrac{1}{2}id^{2}\right)+\tfrac{1}{4}\pi(1-2\operatorname{sign% }\left(k\right))}}
\theta_{0} = \tfrac{3}{2}d^{2}\ln@@{2}+\phase@@{\EulerGamma@{1-\tfrac{1}{2}id^{2}}}+\tfrac{1}{4}\pi(1-2\sign@{k})		 ℜ ⁡ ( 1 - 1 2 ⁢ i ⁢ d 2 ) > 0 1 1 2 imaginary-unit superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(1-\tfrac{1}{2}\mathrm{i}d^{2})>0}} 
theta[0] = (3)/(2)*(d)^(2)* ln(2)+ argument(GAMMA(1 -(1)/(2)*I*(d)^(2)))+(1)/(4)*Pi*(1 - 2*signum(k))

Subscript[\[Theta], 0] == Divide[3,2]*(d)^(2)* Log[2]+ Arg[Gamma[1 -Divide[1,2]*I*(d)^(2)]]+Divide[1,4]*Pi*(1 - 2*Sign[k])
Failure Failure
Failed [300 / 300]
Result: 1.126938891-.4004246008*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, theta[0] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: 1.126938891-.4004246008*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, theta[0] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.1269388909194178, -0.4004246003897078]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[θ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1269388909194178, -0.4004246003897078]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[θ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.11.E14 ϕ ⁢ ( x ) = 2 3 ⁢ | x | 3 / 2 + 3 4 ⁢ d 2 ⁢ ln ⁡ | x | italic-ϕ 𝑥 2 3 superscript 𝑥 3 2 3 4 superscript 𝑑 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{2}{3}|x|^{3/2}+\tfrac{3}{4}d^{2}\ln% |x|}}
\phi(x) = \tfrac{2}{3}|x|^{3/2}+\tfrac{3}{4}d^{2}\ln@@{|x|}		 

phi(x) = (2)/(3)*(abs(x))^(3/2)+(3)/(4)*(d)^(2)* ln(abs(x))

\[Phi][x] == Divide[2,3]*(Abs[x])^(3/2)+Divide[3,4]*(d)^(2)* Log[Abs[x]]
Failure Failure
Failed [300 / 300]
Result: -.777561816e-1+.4866426869*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .4572406346+.7002123003*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.0777561812554926, 0.48664268702259433]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.1267942869321503, 1.0356807926992524]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E15 χ + 3 2 ⁢ d 2 ⁢ ln ⁡ 2 - 1 4 ⁢ π - ph ⁡ Γ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) = n ⁢ π 𝜒 3 2 superscript 𝑑 2 2 1 4 𝜋 phase Euler-Gamma 1 2 𝑖 superscript 𝑑 2 𝑛 𝜋 {\displaystyle{\displaystyle\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-% \operatorname{ph}\Gamma\left(\tfrac{1}{2}id^{2}\right)=n\pi}}
\chi+\tfrac{3}{2}d^{2}\ln@@{2}-\tfrac{1}{4}\pi-\phase@@{\EulerGamma@{\tfrac{1}{2}id^{2}}} = n\pi		 ℜ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) > 0 1 2 imaginary-unit superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}\mathrm{i}d^{2})>0}} 
chi +(3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi - argument(GAMMA((1)/(2)*I*(d)^(2))) = n*Pi

\[Chi]+Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi - Arg[Gamma[Divide[1,2]*I*(d)^(2)]] == n*Pi
Failure Failure
Failed [60 / 60]
Result: -4.109048867-.4004246008*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), n = 1}


Result: -7.250641521-.4004246008*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-4.10904886506357, -0.4004246003897078]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[n, 1], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-7.250641518653364, -0.4004246003897078]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[n, 2], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.11.E17 d 2 = π - 1 ⁢ ln ⁡ ( 1 + k 2 ) superscript 𝑑 2 superscript 𝜋 1 1 superscript 𝑘 2 {\displaystyle{\displaystyle d^{2}=\pi^{-1}\ln\left(1+k^{2}\right)}}
d^{2} = \pi^{-1}\ln@{1+k^{2}}		 sign ⁡ ( k ) = ( - 1 ) n sign 𝑘 superscript 1 𝑛 {\displaystyle{\displaystyle\operatorname{sign}\left(k\right)=(-1)^{n}}} 
(d)^(2) = (Pi)^(- 1)* ln(1 + (k)^(2))

(d)^(2) == (Pi)^(- 1)* Log[1 + (k)^(2)]
Failure Failure
Failed [30 / 30]
Result: .2793644003+.8660254040*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.122999981e-1+.8660254040*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.2793643998473485, 0.8660254037844386]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1]}


Result: Complex[-0.012299998726776007, 0.8660254037844386]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 2]}

... skip entries to safe data
32.11.E18 χ + 3 2 ⁢ d 2 ⁢ ln ⁡ 2 - 1 4 ⁢ π - ph ⁡ Γ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) ≠ n ⁢ π 𝜒 3 2 superscript 𝑑 2 2 1 4 𝜋 phase Euler-Gamma 1 2 𝑖 superscript 𝑑 2 𝑛 𝜋 {\displaystyle{\displaystyle\chi+\tfrac{3}{2}d^{2}\ln 2-\tfrac{1}{4}\pi-% \operatorname{ph}\Gamma\left(\tfrac{1}{2}id^{2}\right)\neq n\pi}}
\chi+\tfrac{3}{2}d^{2}\ln@@{2}-\tfrac{1}{4}\pi-\phase@@{\EulerGamma@{\tfrac{1}{2}id^{2}}} \neq n\pi		 ℜ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) > 0 1 2 imaginary-unit superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}\mathrm{i}d^{2})>0}} 
chi +(3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi - argument(GAMMA((1)/(2)*I*(d)^(2))) <> n*Pi

\[Chi]+Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi - Arg[Gamma[Divide[1,2]*I*(d)^(2)]] \[NotEqual]n*Pi
Failure Failure Successful [Tested: 60]
Failed [60 / 60]
Result: Plus[Complex[-0.4392331450329683, -0.4004246003897078], Times[-3.141592653589793, StringJoin[0.5282230664408086, 1.0]]]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[n, 1], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.4392331450329683, -0.4004246003897078], Times[-3.141592653589793, StringJoin[0.5282230664408086, 2.0]]]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[n, 2], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.11.E20 ψ ⁢ ( x ) = 2 3 ⁢ 2 ⁢ x 3 / 2 - 3 2 ⁢ ρ 2 ⁢ ln ⁡ x 𝜓 𝑥 2 3 2 superscript 𝑥 3 2 3 2 superscript 𝜌 2 𝑥 {\displaystyle{\displaystyle\psi(x)=\tfrac{2}{3}\sqrt{2}x^{3/2}-\tfrac{3}{2}% \rho^{2}\ln x}}
\psi(x) = \tfrac{2}{3}\sqrt{2}x^{3/2}-\tfrac{3}{2}\rho^{2}\ln@@{x}		 

psi(x) = (2)/(3)*sqrt(2)*(x)^(3/2)-(3)/(2)*(rho)^(2)* ln(x)

\[Psi][x] == Divide[2,3]*Sqrt[2]*(x)^(3/2)-Divide[3,2]*\[Rho]^(2)* Log[x]
Failure Failure
Failed [300 / 300]
Result: -.1289138697+1.276714626*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -.4201810172-.6504246006*I
Test Values: {psi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.12891387081109584, 1.276714625954811]
Test Values: {Rule[x, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.1779519764877535, 1.8257527316314692]
Test Values: {Rule[x, 1.5], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E21 σ = - sign ⁡ ( ℑ ⁡ s ) 𝜎 sign 𝑠 {\displaystyle{\displaystyle\sigma=-\operatorname{sign}\left(\Im s\right)}}
\sigma = -\sign@{\imagpart@@{s}}		 ℜ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) > 0 1 2 imaginary-unit superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}\mathrm{i}d^{2})>0}} 
sigma = - signum(Im((exp(Pi*(d)^(2))- 1)^(1/2)* exp(I*((3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi + chi - argument(GAMMA((1)/(2)*I*(d)^(2)))))))

\[Sigma] == - Sign[Im[(Exp[Pi*(d)^(2)]- 1)^(1/2)* Exp[I*(Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi + \[Chi]- Arg[Gamma[Divide[1,2]*I*(d)^(2)]])]]]
Failure Failure
Failed [200 / 200]
Result: -.1339745960+.5000000000*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), sigma = 1/2*3^(1/2)+1/2*I}


Result: -1.500000000+.8660254040*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), sigma = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [200 / 200]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E22 ρ 2 = π - 1 ⁢ ln ⁡ ( ( 1 + | s | 2 ) / | 2 ⁢ ℑ ⁡ s | ) superscript 𝜌 2 superscript 𝜋 1 1 superscript 𝑠 2 2 𝑠 {\displaystyle{\displaystyle\rho^{2}=\pi^{-1}\ln\left((1+|s|^{2})/|2\Im s|% \right)}}
\rho^{2} = \pi^{-1}\ln@{(1+|s|^{2})/|2\imagpart@@{s}|}		 ℜ ⁡ ( 1 2 ⁢ i ⁢ d 2 ) > 0 1 2 imaginary-unit superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}\mathrm{i}d^{2})>0}} 
(rho)^(2) = (Pi)^(- 1)* ln((1 +(abs((exp(Pi*(d)^(2))- 1)^(1/2)* exp(I*((3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi + chi - argument(GAMMA((1)/(2)*I*(d)^(2)))))))^(2))/abs(2*Im((exp(Pi*(d)^(2))- 1)^(1/2)* exp(I*((3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi + chi - argument(GAMMA((1)/(2)*I*(d)^(2))))))))

\[Rho]^(2) == (Pi)^(- 1)* Log[(1 +(Abs[(Exp[Pi*(d)^(2)]- 1)^(1/2)* Exp[I*(Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi + \[Chi]- Arg[Gamma[Divide[1,2]*I*(d)^(2)]])]])^(2))/Abs[2*Im[(Exp[Pi*(d)^(2)]- 1)^(1/2)* Exp[I*(Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi + \[Chi]- Arg[Gamma[Divide[1,2]*I*(d)^(2)]])]]]]
Failure Failure
Failed [200 / 200]
Result: .2989013521+.8660254040*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), rho = 1/2*3^(1/2)+1/2*I}


Result: -.7010986487-.8660254040*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = -1/2+1/2*I*3^(1/2), rho = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [200 / 200]
Result: Complex[0.2989013519411052, 0.8660254037844386]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.3675975407110632, 0.8660254037844386]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E23 θ = - 3 4 ⁢ π - 7 2 ⁢ ρ 2 ⁢ ln ⁡ 2 + ph ⁡ ( 1 + s 2 ) + ph ⁡ Γ ⁡ ( i ⁢ ρ 2 ) 𝜃 3 4 𝜋 7 2 superscript 𝜌 2 2 phase 1 superscript 𝑠 2 phase Euler-Gamma 𝑖 superscript 𝜌 2 {\displaystyle{\displaystyle\theta=-\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}% +\operatorname{ph}\left(1+s^{2}\right)+\operatorname{ph}\Gamma\left(i\rho^{2}% \right)}}
\theta = -\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}+\phase@{1+s^{2}}+\phase@@{\EulerGamma@{i\rho^{2}}}		 ℜ ⁡ ( i ⁢ ρ 2 ) > 0 imaginary-unit superscript 𝜌 2 0 {\displaystyle{\displaystyle\Re(\mathrm{i}\rho^{2})>0}} 
theta = -(3)/(4)*Pi -(7)/(2)*(rho)^(2)* ln(2)+ argument(1 +((exp(Pi*(d)^(2))- 1)^(1/2)* exp(I*((3)/(2)*(d)^(2)* ln(2)-(1)/(4)*Pi + chi - argument(GAMMA((1)/(2)*I*(d)^(2))))))^(2))+ argument(GAMMA(I*(rho)^(2)))

\[Theta] == -Divide[3,4]*Pi -Divide[7,2]*\[Rho]^(2)* Log[2]+ Arg[1 +((Exp[Pi*(d)^(2)]- 1)^(1/2)* Exp[I*(Divide[3,2]*(d)^(2)* Log[2]-Divide[1,4]*Pi + \[Chi]- Arg[Gamma[Divide[1,2]*I*(d)^(2)]])])^(2)]+ Arg[Gamma[I*\[Rho]^(2)]]
Failure Failure
Failed [300 / 300]
Result: 1.925696688-1.600990735*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = 1/2*3^(1/2)+1/2*I, rho = -1/2+1/2*I*3^(1/2), theta = 1/2*3^(1/2)+1/2*I}


Result: .5596712830-1.234965331*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, d = 1/2*3^(1/2)+1/2*I, rho = -1/2+1/2*I*3^(1/2), theta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6978889663556802, -1.6009907342426515]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.496658442211441, -1.6009907342426515]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E27 σ = ( 2 / π ) ⁢ arcsin ⁡ ( π ⁢ λ ) 𝜎 2 𝜋 𝜋 𝜆 {\displaystyle{\displaystyle\sigma=(2/\pi)\operatorname{arcsin}\left(\pi% \lambda\right)}}
\sigma = (2/\pi)\asin@{\pi\lambda}		 

sigma = (2/Pi)*arcsin(Pi*lambda)

\[Sigma] == (2/Pi)*ArcSin[Pi*\[Lambda]]
Failure Failure
Failed [100 / 100]
Result: .2138525505-.6623078870*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I}


Result: -1.152172854-.2962824830*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, sigma = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [100 / 100]
Result: Complex[0.2138525499640901, -0.6623078873679977]
Test Values: {Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.1521728538203484, -0.296282483583559]
Test Values: {Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E28 B = 2 - 2 ⁢ σ ⁢ Γ 2 ⁡ ( 1 2 ⁢ ( 1 - σ ) ) ⁢ Γ ⁡ ( 1 2 ⁢ ( 1 + σ ) + ν ) Γ 2 ⁡ ( 1 2 ⁢ ( 1 + σ ) ) ⁢ Γ ⁡ ( 1 2 ⁢ ( 1 - σ ) + ν ) 𝐵 superscript 2 2 𝜎 Euler-Gamma 2 1 2 1 𝜎 Euler-Gamma 1 2 1 𝜎 𝜈 Euler-Gamma 2 1 2 1 𝜎 Euler-Gamma 1 2 1 𝜎 𝜈 {\displaystyle{\displaystyle B=2^{-2\sigma}\frac{{\Gamma^{2}}\left(\tfrac{1}{2% }(1-\sigma)\right)\Gamma\left(\tfrac{1}{2}(1+\sigma)+\nu\right)}{{\Gamma^{2}}% \left(\tfrac{1}{2}(1+\sigma)\right)\Gamma\left(\tfrac{1}{2}(1-\sigma)+\nu% \right)}}}
B = 2^{-2\sigma}\frac{\EulerGamma^{2}@{\tfrac{1}{2}(1-\sigma)}\EulerGamma@{\tfrac{1}{2}(1+\sigma)+\nu}}{\EulerGamma^{2}@{\tfrac{1}{2}(1+\sigma)}\EulerGamma@{\tfrac{1}{2}(1-\sigma)+\nu}}		 ℜ ⁡ ( 1 2 ⁢ ( 1 - σ ) ) > 0 , ℜ ⁡ ( 1 2 ⁢ ( 1 + σ ) + ν ) > 0 , ℜ ⁡ ( 1 2 ⁢ ( 1 + σ ) ) > 0 , ℜ ⁡ ( 1 2 ⁢ ( 1 - σ ) + ν ) > 0 formulae-sequence 1 2 1 𝜎 0 formulae-sequence 1 2 1 𝜎 𝜈 0 formulae-sequence 1 2 1 𝜎 0 1 2 1 𝜎 𝜈 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}(1-\sigma))>0,\Re(\tfrac{1}{2}(1+% \sigma)+\nu)>0,\Re(\tfrac{1}{2}(1+\sigma))>0,\Re(\tfrac{1}{2}(1-\sigma)+\nu)>0}} 
B = (2)^(- 2*sigma)*((GAMMA((1)/(2)*(1 - sigma)))^(2)* GAMMA((1)/(2)*(1 + sigma)+ nu))/((GAMMA((1)/(2)*(1 + sigma)))^(2)* GAMMA((1)/(2)*(1 - sigma)+ nu))

B == (2)^(- 2*\[Sigma])*Divide[(Gamma[Divide[1,2]*(1 - \[Sigma])])^(2)* Gamma[Divide[1,2]*(1 + \[Sigma])+ \[Nu]],(Gamma[Divide[1,2]*(1 + \[Sigma])])^(2)* Gamma[Divide[1,2]*(1 - \[Sigma])+ \[Nu]]]
Failure Failure
Failed [300 / 300]
Result: 3.808977659-.2371191295*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I}


Result: .9147008442+.353764288e-1*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, sigma = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[3.808977656026658, -0.23711913260929035]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.914700843688173, 0.035376428936519655]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[σ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E31 h * = 1 / ( π 1 / 2 ⁢ Γ ⁡ ( ν + 1 ) ) superscript ℎ 1 superscript 𝜋 1 2 Euler-Gamma 𝜈 1 {\displaystyle{\displaystyle h^{*}=\ifrac{1}{\left(\pi^{1/2}\Gamma\left(\nu+1% \right)\right)}}}
h^{*} = \ifrac{1}{\left(\pi^{1/2}\EulerGamma@{\nu+1}\right)}		 ℜ ⁡ ( ν + 1 ) > 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0}} 
(h)^(*) = (1)/((Pi)^(1/2)* GAMMA(nu + 1))

(h)^(*) == Divide[1,(Pi)^(1/2)* Gamma[\[Nu]+ 1]]
Error Failure - Error
32.11.E34 ϕ ⁢ ( x ) = 1 3 ⁢ 3 ⁢ x 2 - 4 3 ⁢ d 2 ⁢ 3 ⁢ ln ⁡ ( 2 ⁢ | x | ) italic-ϕ 𝑥 1 3 3 superscript 𝑥 2 4 3 superscript 𝑑 2 3 2 𝑥 {\displaystyle{\displaystyle\phi(x)=\tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2% }\sqrt{3}\ln\left(\sqrt{2}|x|\right)}}
\phi(x) = \tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\ln@{\sqrt{2}|x|}		 

phi(x) = (1)/(3)*sqrt(3)*(x)^(2)-(4)/(3)*(d)^(2)*sqrt(3)*ln(sqrt(2)*abs(x))
 
\[Phi][x] == Divide[1,3]*Sqrt[3]*(x)^(2)-Divide[4,3]*(d)^(2)*Sqrt[3]*Log[Sqrt[2]*Abs[x]]
 Failure Failure
Failed [300 / 300]
Result: .8683794902+2.254077396*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -.1115135772-.4431471813*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.8683794899108137, 2.2540773967762746]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-1.180658615765844, 2.8031155024529326]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E35 d 2 = - 1 4 ⁢ 3 ⁢ π - 1 ⁢ ln ⁡ ( 1 - | μ | 2 ) superscript 𝑑 2 1 4 3 superscript 𝜋 1 1 superscript 𝜇 2 {\displaystyle{\displaystyle d^{2}=-\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln\left(1-|% \mu|^{2}\right)}}
d^{2} = -\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln@{1-|\mu|^{2}}		 

(d)^(2) = -(1)/(4)*sqrt(3)*(Pi)^(- 1)* ln(1 -(abs(mu))^(2))
 
(d)^(2) == -Divide[1,4]*Sqrt[3]*(Pi)^(- 1)* Log[1 -(Abs[\[Mu]])^(2)]
 Failure Failure
Failed [100 / 100]
Result: Float(-infinity)+.8660254040*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I}

Result: Float(-infinity)+.8660254040*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, mu = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [100 / 100]
Result: DirectedInfinity[-1]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: DirectedInfinity[-1]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E36 θ 0 = 1 3 ⁢ d 2 ⁢ 3 ⁢ ln ⁡ 3 + 2 3 ⁢ π ⁢ ν + 7 12 ⁢ π + ph ⁡ μ + ph ⁡ Γ ⁡ ( - 2 3 ⁢ i ⁢ 3 ⁢ d 2 ) subscript 𝜃 0 1 3 superscript 𝑑 2 3 3 2 3 𝜋 𝜈 7 12 𝜋 phase 𝜇 phase Euler-Gamma 2 3 𝑖 3 superscript 𝑑 2 {\displaystyle{\displaystyle\theta_{0}=\tfrac{1}{3}d^{2}\sqrt{3}\ln 3+\tfrac{2% }{3}\pi\nu+\tfrac{7}{12}\pi+\operatorname{ph}\mu+\operatorname{ph}\Gamma\left(% -\tfrac{2}{3}i\sqrt{3}d^{2}\right)}}
\theta_{0} = \tfrac{1}{3}d^{2}\sqrt{3}\ln@@{3}+\tfrac{2}{3}\pi\nu+\tfrac{7}{12}\pi+\phase@@{\mu}+\phase@@{\EulerGamma@{-\tfrac{2}{3}i\sqrt{3}d^{2}}}		 ℜ ⁡ ( - 2 3 ⁢ i ⁢ 3 ⁢ d 2 ) > 0 2 3 imaginary-unit 3 superscript 𝑑 2 0 {\displaystyle{\displaystyle\Re(-\tfrac{2}{3}\mathrm{i}\sqrt{3}d^{2})>0}} 
theta[0] = (1)/(3)*(d)^(2)*sqrt(3)*ln(3)+(2)/(3)*Pi*nu +(7)/(12)*Pi + argument(mu)+ argument(GAMMA(-(2)/(3)*I*sqrt(3)*(d)^(2)))
 
Subscript[\[Theta], 0] == Divide[1,3]*(d)^(2)*Sqrt[3]*Log[3]+Divide[2,3]*Pi*\[Nu]+Divide[7,12]*Pi + Arg[\[Mu]]+ Arg[Gamma[-Divide[2,3]*I*Sqrt[3]*(d)^(2)]]
 Failure Failure
Failed [300 / 300]
Result: -3.888102442-1.096503697*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, theta[0] = 1/2*3^(1/2)+1/2*I}

Result: -5.254127846-.7304782927*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, theta[0] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-3.888102439563878, -1.0965036955306524]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[θ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-5.254127843348316, -0.7304782917462136]
Test Values: {Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[θ, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.11.E37 μ = 1 + ( 2 ⁢ i ⁢ h ⁢ π 3 / 2 ⁢ exp ⁡ ( - i ⁢ π ⁢ ν ) / Γ ⁡ ( - ν ) ) 𝜇 1 2 𝑖 ℎ superscript 𝜋 3 2 𝑖 𝜋 𝜈 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\mu=1+\left(\ifrac{2ih\pi^{3/2}\exp\left(-i\pi\nu% \right)}{\Gamma\left(-\nu\right)}\right)}}
\mu = 1+\left(\ifrac{2ih\pi^{3/2}\exp@{-i\pi\nu}}{\EulerGamma@{-\nu}}\right)		 ℜ ⁡ ( - ν ) > 0 𝜈 0 {\displaystyle{\displaystyle\Re(-\nu)>0}} 
mu = 1 +((2*I*h*(Pi)^(3/2)* exp(- I*Pi*nu))/(GAMMA(- nu)))
 
\[Mu] == 1 +(Divide[2*I*h*(Pi)^(3/2)* Exp[- I*Pi*\[Nu]],Gamma[- \[Nu]]])
 Failure Failure
Failed [300 / 300]
Result: 241.2310915-105.5149067*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2)}

Result: -1.289758519+2.890481636*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[241.23109103950634, -105.514906477147]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-1.289758518042884, 2.89048163412207]
Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
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