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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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|} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d^{2} = -\pi^{-1}\ln@{1-k^{2}}}
d^{2} = -\pi^{-1}\ln@{1-k^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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})}
\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}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(1-\tfrac{1}{2}\iunit 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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|} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \chi+\tfrac{3}{2}d^{2}\ln@@{2}-\tfrac{1}{4}\pi-\phase@@{\EulerGamma@{\tfrac{1}{2}id^{2}}} = n\pi}
\chi+\tfrac{3}{2}d^{2}\ln@@{2}-\tfrac{1}{4}\pi-\phase@@{\EulerGamma@{\tfrac{1}{2}id^{2}}} = n\pi |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}\iunit 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d^{2} = \pi^{-1}\ln@{1+k^{2}}}
d^{2} = \pi^{-1}\ln@{1+k^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sign@{k} = (-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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \chi+\tfrac{3}{2}d^{2}\ln@@{2}-\tfrac{1}{4}\pi-\phase@@{\EulerGamma@{\tfrac{1}{2}id^{2}}} \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 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}\iunit 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sigma = -\sign@{\imagpart@@{s}}}
\sigma = -\sign@{\imagpart@@{s}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}\iunit 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \rho^{2} = \pi^{-1}\ln@{(1+|s|^{2})/|2\imagpart@@{s}|}}
\rho^{2} = \pi^{-1}\ln@{(1+|s|^{2})/|2\imagpart@@{s}|} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}\iunit 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \theta = -\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}+\phase@{1+s^{2}}+\phase@@{\EulerGamma@{i\rho^{2}}}}
\theta = -\tfrac{3}{4}\pi-\tfrac{7}{2}\rho^{2}\ln{2}+\phase@{1+s^{2}}+\phase@@{\EulerGamma@{i\rho^{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\iunit \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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sigma = (2/\pi)\asin@{\pi\lambda}}
\sigma = (2/\pi)\asin@{\pi\lambda} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 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}}}
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}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}(1-\sigma))} > 0, \realpart@@{(\tfrac{1}{2}(1+\sigma)+\nu)} > 0, \realpart@@{(\tfrac{1}{2}(1+\sigma))} > 0, \realpart@@{(\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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h^{*} = \ifrac{1}{\left(\pi^{1/2}\EulerGamma@{\nu+1}\right)}}
h^{*} = \ifrac{1}{\left(\pi^{1/2}\EulerGamma@{\nu+1}\right)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi(x) = \tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\ln@{\sqrt{2}|x|}}
\phi(x) = \tfrac{1}{3}\sqrt{3}x^{2}-\tfrac{4}{3}d^{2}\sqrt{3}\ln@{\sqrt{2}|x|} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d^{2} = -\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln@{1-|\mu|^{2}}}
d^{2} = -\tfrac{1}{4}\sqrt{3}\pi^{-1}\ln@{1-|\mu|^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(-\tfrac{2}{3}\iunit \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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = 1+\left(\ifrac{2ih\pi^{3/2}\exp@{-i\pi\nu}}{\EulerGamma@{-\nu}}\right)}
\mu = 1+\left(\ifrac{2ih\pi^{3/2}\exp@{-i\pi\nu}}{\EulerGamma@{-\nu}}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(-\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 |