Integrals with Coalescing Saddles - 36.4 Bifurcation Sets
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 36.4.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = 0}
x = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = 0 |
x == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 27x^{2} = -8y^{3}}
27x^{2} = -8y^{3} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 27*(x)^(2) = - 8*(y)^(3) |
27*(x)^(2) == - 8*(y)^(3) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = 3t^{2}(z+5t^{2})}
x = 3t^{2}(z+5t^{2}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = 3*(t)^(2)*((x + y*I)+ 5*(t)^(2)) |
x == 3*(t)^(2)*((x + y*I)+ 5*(t)^(2)) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = -t(3z+10t^{2})}
y = -t(3z+10t^{2}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\infty < t, t < \infty} | y = - t*(3*(x + y*I)+ 10*(t)^(2)) |
y == - t*(3*(x + y*I)+ 10*(t)^(2)) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = 0}
y = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | y = 0 |
y == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z \leq 0}
z \leq 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | z <= 0 |
z <= 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = \tfrac{9}{20}z^{2}}
x = \tfrac{9}{20}z^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = (9)/(20)*(x + y*I)^(2) |
x == Divide[9,20]*(x + y*I)^(2) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z \leq 0}
z \leq 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | z <= 0 |
z <= 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = -\tfrac{3}{20}z^{2}}
x = -\tfrac{3}{20}z^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = -(3)/(20)*(x + y*I)^(2) |
x == -Divide[3,20]*(x + y*I)^(2) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 10y^{2} = -4z^{3}}
10y^{2} = -4z^{3} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 10*(y)^(2) = - 4*(x + y*I)^(3) |
10*(y)^(2) == - 4*(x + y*I)^(3) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.4#Ex11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = \tfrac{1}{3}z^{2}(-\cos@{2\phi}-2\cos@@{\phi})}
x = \tfrac{1}{3}z^{2}(-\cos@{2\phi}-2\cos@@{\phi}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = (1)/(3)*(x + y*I)^(2)*(- cos(2*phi)- 2*cos(phi))
|
x == Divide[1,3]*(x + y*I)^(2)*(- Cos[2*\[Phi]]- 2*Cos[\[Phi]])
|
Failure | Failure | Failed [180 / 180] Result: -1.430780988-1.820001528*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}
Result: 4.430780988+1.820001528*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [180 / 180]
Result: Complex[-1.4307809877134479, -1.8200015289703178]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[6.36271355956067, -6.045425917929804]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 36.4#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = \tfrac{1}{3}z^{2}(\sin@{2\phi}-2\sin@@{\phi})}
y = \tfrac{1}{3}z^{2}(\sin@{2\phi}-2\sin@@{\phi}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \phi, \phi \leq 2\pi} | y = (1)/(3)*(x + y*I)^(2)*(sin(2*phi)- 2*sin(phi))
|
y == Divide[1,3]*(x + y*I)^(2)*(Sin[2*\[Phi]]- 2*Sin[\[Phi]])
|
Failure | Failure | Failed [54 / 54] Result: -1.500000000-2.780804948*I
Test Values: {phi = 3/2, x = 3/2, y = -3/2}
Result: 1.500000000+2.780804948*I
Test Values: {phi = 3/2, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [54 / 54]
Result: Complex[-1.5, -2.7808049477223626]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, 1.5]}
Result: Complex[-1.5, -0.17607013860076426]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, 0.5]}
... skip entries to safe data |
| 36.4.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x+iy = -z^{2}\exp@{\tfrac{2}{3}i\pi m}}
x+iy = -z^{2}\exp@{\tfrac{2}{3}i\pi m} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x + I*y = -(x + y*I)^(2)* exp((2)/(3)*I*Pi*m)
|
x + I*y == -(x + y*I)^(2)* Exp[Divide[2,3]*I*Pi*m]
|
Error | Failure | - | Skip - No test values generated |
| 36.4#Ex13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}+ 2\exp@{-\tau})}
x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}+ 2\exp@{-\tau}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = -(1)/(12)*(x + y*I)^(2)*(exp(2*tau)+ 2*exp(- tau))
|
x == -Divide[1,12]*(x + y*I)^(2)*(Exp[2*\[Tau]]+ 2*Exp[- \[Tau]])
|
Failure | Failure | Failed [180 / 180] Result: 3.132329502-1.422064683*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}
Result: -.132329502+1.422064683*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [180 / 180]
Result: Complex[3.1323295016161525, -1.4220646828050367]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.694217640570763, -0.7789550680773418]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 36.4#Ex13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}- 2\exp@{-\tau})}
x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}- 2\exp@{-\tau}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = -(1)/(12)*(x + y*I)^(2)*(exp(2*tau)- 2*exp(- tau))
|
x == -Divide[1,12]*(x + y*I)^(2)*(Exp[2*\[Tau]]- 2*Exp[- \[Tau]])
|
Failure | Failure | Failed [180 / 180] Result: 3.434813476-.8683714834*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}
Result: -.434813476+.8683714834*I
Test Values: {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [180 / 180]
Result: Complex[3.4348134754242143, -0.8683714827345889]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[2.578112467364518, 0.8232541553428006]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 36.4#Ex14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}+ 2\exp@{\tau})}
y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}+ 2\exp@{\tau}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\infty \leq \tau, \tau < \infty.} | y = -(1)/(12)*(x + y*I)^(2)*(exp(- 2*tau)+ 2*exp(tau))
|
y == -Divide[1,12]*(x + y*I)^(2)*(Exp[- 2*\[Tau]]+ 2*Exp[\[Tau]])
|
Failure | Failure | Failed [108 / 108] Result: -1.500000000-7.699423965*I
Test Values: {tau = -3/2, x = 3/2, y = -3/2}
Result: 1.500000000+7.699423965*I
Test Values: {tau = -3/2, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [108 / 108]
Result: Complex[-1.5, -7.699423966306697]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, -1.5]}
Result: Complex[-1.5, -3.3799369533914976]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, 1.5]}
... skip entries to safe data |
| 36.4#Ex14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}- 2\exp@{\tau})}
y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}- 2\exp@{\tau}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\infty \leq \tau, \tau < \infty.} | y = -(1)/(12)*(x + y*I)^(2)*(exp(- 2*tau)- 2*exp(tau))
|
y == -Divide[1,12]*(x + y*I)^(2)*(Exp[- 2*\[Tau]]- 2*Exp[\[Tau]])
|
Failure | Failure | Failed [108 / 108] Result: -1.500000000-7.364728725*I
Test Values: {tau = -3/2, x = 3/2, y = -3/2}
Result: 1.500000000+7.364728725*I
Test Values: {tau = -3/2, x = 3/2, y = 3/2}
... skip entries to safe data |
Failed [108 / 108]
Result: Complex[-1.5, -7.364728726084054]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, -1.5]}
Result: Complex[-1.5, 3.3425966521155988]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, 1.5]}
... skip entries to safe data |
| 36.4.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = y}
x = y |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x = y |
x == y |
Skipped - no semantic math | Skipped - no semantic math | - | - |