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 x = 0 π‘₯ 0 {\displaystyle{\displaystyle x=0}}
x = 0		 

x = 0
 
x == 0
 Skipped - no semantic math Skipped - no semantic math - -
36.4.E6 27 ⁒ x 2 = - 8 ⁒ y 3 27 superscript π‘₯ 2 8 superscript 𝑦 3 {\displaystyle{\displaystyle 27x^{2}=-8y^{3}}}
27x^{2} = -8y^{3}		 

27*(x)^(2) = - 8*(y)^(3)
 
27*(x)^(2) == - 8*(y)^(3)
 Skipped - no semantic math Skipped - no semantic math - -
36.4#Ex3 x = 3 ⁒ t 2 ⁒ ( z + 5 ⁒ t 2 ) π‘₯ 3 superscript 𝑑 2 𝑧 5 superscript 𝑑 2 {\displaystyle{\displaystyle x=3t^{2}(z+5t^{2})}}
x = 3t^{2}(z+5t^{2})		 

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 y = - t ⁒ ( 3 ⁒ z + 10 ⁒ t 2 ) 𝑦 𝑑 3 𝑧 10 superscript 𝑑 2 {\displaystyle{\displaystyle y=-t(3z+10t^{2})}}
y = -t(3z+10t^{2})		 - ∞ < t , t < ∞ formulae-sequence 𝑑 𝑑 {\displaystyle{\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 y = 0 𝑦 0 {\displaystyle{\displaystyle y=0}}
y = 0		 

y = 0
 
y == 0
 Skipped - no semantic math Skipped - no semantic math - -
36.4#Ex6 z ≀ 0 𝑧 0 {\displaystyle{\displaystyle z\leq 0}}
z \leq 0		 

z <= 0
 
z <= 0
 Skipped - no semantic math Skipped - no semantic math - -
36.4#Ex7 x = 9 20 ⁒ z 2 π‘₯ 9 20 superscript 𝑧 2 {\displaystyle{\displaystyle x=\tfrac{9}{20}z^{2}}}
x = \tfrac{9}{20}z^{2}		 

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 z ≀ 0 𝑧 0 {\displaystyle{\displaystyle z\leq 0}}
z \leq 0		 

z <= 0
 
z <= 0
 Skipped - no semantic math Skipped - no semantic math - -
36.4#Ex9 x = - 3 20 ⁒ z 2 π‘₯ 3 20 superscript 𝑧 2 {\displaystyle{\displaystyle x=-\tfrac{3}{20}z^{2}}}
x = -\tfrac{3}{20}z^{2}		 

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 10 ⁒ y 2 = - 4 ⁒ z 3 10 superscript 𝑦 2 4 superscript 𝑧 3 {\displaystyle{\displaystyle 10y^{2}=-4z^{3}}}
10y^{2} = -4z^{3}		 

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 x = 1 3 ⁒ z 2 ⁒ ( - cos ⁑ ( 2 ⁒ Ο• ) - 2 ⁒ cos ⁑ Ο• ) π‘₯ 1 3 superscript 𝑧 2 2 italic-Ο• 2 italic-Ο• {\displaystyle{\displaystyle x=\tfrac{1}{3}z^{2}(-\cos\left(2\phi\right)-2\cos% \phi)}}
x = \tfrac{1}{3}z^{2}(-\cos@{2\phi}-2\cos@@{\phi})		 

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[Ο•, 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[Ο•, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
36.4#Ex12 y = 1 3 ⁒ z 2 ⁒ ( sin ⁑ ( 2 ⁒ Ο• ) - 2 ⁒ sin ⁑ Ο• ) 𝑦 1 3 superscript 𝑧 2 2 italic-Ο• 2 italic-Ο• {\displaystyle{\displaystyle y=\tfrac{1}{3}z^{2}(\sin\left(2\phi\right)-2\sin% \phi)}}
y = \tfrac{1}{3}z^{2}(\sin@{2\phi}-2\sin@@{\phi})		 0 ≀ Ο• , Ο• ≀ 2 ⁒ Ο€ formulae-sequence 0 italic-Ο• italic-Ο• 2 πœ‹ {\displaystyle{\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[Ο•, 1.5]}


Result: Complex[-1.5, -0.17607013860076426]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο•, 0.5]}

... skip entries to safe data
36.4.E11 x + i ⁒ y = - z 2 ⁒ exp ⁑ ( 2 3 ⁒ i ⁒ Ο€ ⁒ m ) π‘₯ 𝑖 𝑦 superscript 𝑧 2 2 3 𝑖 πœ‹ π‘š {\displaystyle{\displaystyle x+iy=-z^{2}\exp\left(\tfrac{2}{3}i\pi m\right)}}
x+iy = -z^{2}\exp@{\tfrac{2}{3}i\pi m}		 

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 x = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( 2 ⁒ Ο„ ) + 2 ⁒ exp ⁑ ( - Ο„ ) ) π‘₯ 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle x=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)+2% \exp\left(-\tau\right))}}
x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}+ 2\exp@{-\tau})		 

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[Ο„, 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[Ο„, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
36.4#Ex13 x = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( 2 ⁒ Ο„ ) - 2 ⁒ exp ⁑ ( - Ο„ ) ) π‘₯ 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle x=-\tfrac{1}{12}z^{2}(\exp\left(2\tau\right)-2% \exp\left(-\tau\right))}}
x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}- 2\exp@{-\tau})		 

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[Ο„, 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[Ο„, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
36.4#Ex14 y = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( - 2 ⁒ Ο„ ) + 2 ⁒ exp ⁑ ( Ο„ ) ) 𝑦 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle y=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)+2% \exp\left(\tau\right))}}
y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}+ 2\exp@{\tau})		 - ∞ ≀ Ο„ , Ο„ < ∞ . formulae-sequence 𝜏 𝜏 {\displaystyle{\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[Ο„, -1.5]}


Result: Complex[-1.5, -3.3799369533914976]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο„, 1.5]}

... skip entries to safe data
36.4#Ex14 y = - 1 12 ⁒ z 2 ⁒ ( exp ⁑ ( - 2 ⁒ Ο„ ) - 2 ⁒ exp ⁑ ( Ο„ ) ) 𝑦 1 12 superscript 𝑧 2 2 𝜏 2 𝜏 {\displaystyle{\displaystyle y=-\tfrac{1}{12}z^{2}(\exp\left(-2\tau\right)-2% \exp\left(\tau\right))}}
y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}- 2\exp@{\tau})		 - ∞ ≀ Ο„ , Ο„ < ∞ . formulae-sequence 𝜏 𝜏 {\displaystyle{\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[Ο„, -1.5]}


Result: Complex[-1.5, 3.3425966521155988]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Ο„, 1.5]}

... skip entries to safe data
36.4.E13 x = y π‘₯ 𝑦 {\displaystyle{\displaystyle x=y}}
x = y		 

x = y
 
x == y
 Skipped - no semantic math Skipped - no semantic math - -
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