Integrals with Coalescing Saddles - 36.5 Stokes Sets

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
36.5.E2 y 3 = 27 4 ⁒ ( 27 - 5 ) ⁒ x 2 superscript 𝑦 3 27 4 27 5 superscript π‘₯ 2 {\displaystyle{\displaystyle y^{3}=\tfrac{27}{4}\left(\sqrt{27}-5\right)x^{2}}}
y^{3} = \tfrac{27}{4}\left(\sqrt{27}-5\right)x^{2}		 

(y)^(3) = (27)/(4)*(sqrt(27)- 5)*(x)^(2)
 
(y)^(3) == Divide[27,4]*(Sqrt[27]- 5)*(x)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
36.5#Ex3 x = B + ⁒ | y | 4 / 3 π‘₯ subscript 𝐡 superscript 𝑦 4 3 {\displaystyle{\displaystyle x=B_{+}|y|^{4/3}}}
x = B_{+}|y|^{4/3}		 

x = B[+]*(abs(y))^(4/3)
 
x == Subscript[B, +]*(Abs[y])^(4/3)
 Skipped - no semantic math Skipped - no semantic math - -
36.5#Ex4 B + = 10 - 1 / 3 ⁒ ( 2 ⁒ x + 4 / 3 - 1 2 ⁒ x + - 2 / 3 ) subscript 𝐡 superscript 10 1 3 2 superscript subscript π‘₯ 4 3 1 2 superscript subscript π‘₯ 2 3 {\displaystyle{\displaystyle B_{+}=10^{-1/3}\left(2x_{+}^{4/3}-\tfrac{1}{2}x_{% +}^{-2/3}\right)}}
B_{+} = 10^{-1/3}\left(2x_{+}^{4/3}-\tfrac{1}{2}x_{+}^{-2/3}\right)		 

B[+] = (10)^(- 1/3)*(2*(x[+])^(4/3)-(1)/(2)*(x[+])^(- 2/3))
 
Subscript[B, +] == (10)^(- 1/3)*(2*(Subscript[x, +])^(4/3)-Divide[1,2]*(Subscript[x, +])^(- 2/3))
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E4 80 ⁒ x 5 - 40 ⁒ x 4 - 55 ⁒ x 3 + 5 ⁒ x 2 + 20 ⁒ x - 1 = 0 80 superscript π‘₯ 5 40 superscript π‘₯ 4 55 superscript π‘₯ 3 5 superscript π‘₯ 2 20 π‘₯ 1 0 {\displaystyle{\displaystyle 80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0}}
80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1 = 0		 

80*(x)^(5)- 40*(x)^(4)- 55*(x)^(3)+ 5*(x)^(2)+ 20*x - 1 = 0
 
80*(x)^(5)- 40*(x)^(4)- 55*(x)^(3)+ 5*(x)^(2)+ 20*x - 1 == 0
 Skipped - no semantic math Skipped - no semantic math - -
36.5#Ex5 B - = - 1.69916 subscript 𝐡 1.69916 {\displaystyle{\displaystyle B_{-}=-1.69916}}
B_{-} = -1.69916		 

B[-] = - 1.69916
 
Subscript[B, -] == - 1.69916
 Skipped - no semantic math Skipped - no semantic math - -
36.5#Ex6 B + = 0.33912 subscript 𝐡 0.33912 {\displaystyle{\displaystyle B_{+}=0.33912}}
B_{+} = 0.33912		 

B[+] = 0.33912
 
Subscript[B, +] == 0.33912
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E7 X = 9 20 + 20 ⁒ u 4 - Y 2 20 ⁒ u 2 + 6 ⁒ u 2 ⁒ sign ⁑ ( z ) 𝑋 9 20 20 superscript 𝑒 4 superscript π‘Œ 2 20 superscript 𝑒 2 6 superscript 𝑒 2 sign 𝑧 {\displaystyle{\displaystyle X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^% {2}\operatorname{sign}\left(z\right)}}
X = \dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\sign@{z}		 

X = (9)/(20)+ 20*(u)^(4)-((y/(abs(x + y*I))^(3/2))^(2))/(20*(u)^(2))+ 6*(u)^(2)* signum(x + y*I)

X == Divide[9,20]+ 20*(u)^(4)-Divide[(y/(Abs[x + y*I])^(3/2))^(2),20*(u)^(2)]+ 6*(u)^(2)* Sign[x + y*I]
Failure Failure
Failed [300 / 300]
Result: 4.626363000-18.38362857*I
Test Values: {X = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}


Result: 11.97483223-22.62626926*I
Test Values: {X = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[4.626363002559913, -18.383628553565494]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[X, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}


Result: Complex[11.974832230909447, -22.62626924068478]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[X, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, 1.5]}

... skip entries to safe data
36.5.E8 16 ⁒ u 5 - Y 2 10 ⁒ u + 4 ⁒ u 3 ⁒ sign ⁑ ( z ) - 3 10 ⁒ | Y | ⁒ sign ⁑ ( z ) + 4 ⁒ t 5 + 2 ⁒ t 3 ⁒ sign ⁑ ( z ) + | Y | ⁒ t 2 = 0 16 superscript 𝑒 5 superscript π‘Œ 2 10 𝑒 4 superscript 𝑒 3 sign 𝑧 3 10 π‘Œ sign 𝑧 4 superscript 𝑑 5 2 superscript 𝑑 3 sign 𝑧 π‘Œ superscript 𝑑 2 0 {\displaystyle{\displaystyle 16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\operatorname{% sign}\left(z\right)-\frac{3}{10}|Y|\operatorname{sign}\left(z\right)+4t^{5}+2t% ^{3}\operatorname{sign}\left(z\right)+|Y|t^{2}=0}}
16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\sign@{z}-\frac{3}{10}|Y|\sign@{z}+4t^{5}+2t^{3}\sign@{z}+|Y|t^{2} = 0		 

16*(u)^(5)-((y/(abs(x + y*I))^(3/2))^(2))/(10*u)+ 4*(u)^(3)* signum(x + y*I)-(3)/(10)*abs(y/(abs(x + y*I))^(3/2))*signum(x + y*I)+ 4*(t)^(5)+ 2*(t)^(3)* signum(x + y*I)+abs(y/(abs(x + y*I))^(3/2))*(t)^(2) = 0

16*(u)^(5)-Divide[(y/(Abs[x + y*I])^(3/2))^(2),10*u]+ 4*(u)^(3)* Sign[x + y*I]-Divide[3,10]*Abs[y/(Abs[x + y*I])^(3/2)]*Sign[x + y*I]+ 4*(t)^(5)+ 2*(t)^(3)* Sign[x + y*I]+Abs[y/(Abs[x + y*I])^(3/2)]*(t)^(2) == 0
Failure Failure
Failed [300 / 300]
Result: -45.20699439+15.71617138*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}


Result: -50.86384865+5.964253129*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-45.20699439419109, 15.716171367970516]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[-50.863848643683475, 5.964253107561413]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
36.5.E9 t = - u + ( | Y | 10 ⁒ u - u 2 - 3 10 ⁒ sign ⁑ ( z ) ) 1 / 2 𝑑 𝑒 superscript π‘Œ 10 𝑒 superscript 𝑒 2 3 10 sign 𝑧 1 2 {\displaystyle{\displaystyle t=-u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}% \operatorname{sign}\left(z\right)\right)^{1/2}}}
t = -u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}\sign@{z}\right)^{1/2}		 

t = - u +((abs(y/(abs(x + y*I))^(3/2)))/(10*u)- (u)^(2)-(3)/(10)*signum(x + y*I))^(1/2)

t == - u +(Divide[Abs[y/(Abs[x + y*I])^(3/2)],10*u]- (u)^(2)-Divide[3,10]*Sign[x + y*I])^(1/2)
Failure Failure
Failed [300 / 300]
Result: -1.010332328+1.400961906*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}


Result: -1.190759425+1.489998263*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.0103323276990124, 1.4009619057655258]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[-1.1907594253432556, 1.489998262392432]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
36.5.E10 160 ⁒ u 6 + 40 ⁒ u 4 = Y 2 160 superscript 𝑒 6 40 superscript 𝑒 4 superscript π‘Œ 2 {\displaystyle{\displaystyle 160u^{6}+40u^{4}=Y^{2}}}
160u^{6}+40u^{4} = Y^{2}		 

160*(u)^(6)+ 40*(u)^(4) = (y/(abs(x + y*I))^(3/2))^(2)
 
160*(u)^(6)+ 40*(u)^(4) == (y/(Abs[x + y*I])^(3/2))^(2)
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E11 x z 2 = - 1 - 12 ⁒ u 2 + 8 ⁒ u - | y z 2 | ⁒ 1 3 - u ( u ⁒ ( 2 3 - u ) ) 1 / 2 π‘₯ superscript 𝑧 2 1 12 superscript 𝑒 2 8 𝑒 𝑦 superscript 𝑧 2 1 3 𝑒 superscript 𝑒 2 3 𝑒 1 2 {\displaystyle{\displaystyle\frac{x}{z^{2}}=-1-12u^{2}+8u-\left|\frac{y}{z^{2}% }\right|\dfrac{\frac{1}{3}-u}{\left(u\left(\frac{2}{3}-u\right)\right)^{1/2}}}}
\frac{x}{z^{2}} = -1-12u^{2}+8u-\left|\frac{y}{z^{2}}\right|\dfrac{\frac{1}{3}-u}{\left(u\left(\frac{2}{3}-u\right)\right)^{1/2}}		 

(x)/((x + y*I)^(2)) = - 1 - 12*(u)^(2)+ 8*u -abs((y)/((x + y*I)^(2)))*((1)/(3)- u)/((u*((2)/(3)- u))^(1/2))
 
Divide[x,(x + y*I)^(2)] == - 1 - 12*(u)^(2)+ 8*u -Abs[Divide[y,(x + y*I)^(2)]]*Divide[Divide[1,3]- u,(u*(Divide[2,3]- u))^(1/2)]
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E12 8 ⁒ u 3 - 4 ⁒ u 2 - | y 3 ⁒ z 2 | ⁒ ( u 2 3 - u ) 1 / 2 = y 2 6 ⁒ w ⁒ z 4 - 2 ⁒ w 3 - 2 ⁒ w 2 8 superscript 𝑒 3 4 superscript 𝑒 2 𝑦 3 superscript 𝑧 2 superscript 𝑒 2 3 𝑒 1 2 superscript 𝑦 2 6 𝑀 superscript 𝑧 4 2 superscript 𝑀 3 2 superscript 𝑀 2 {\displaystyle{\displaystyle 8u^{3}-4u^{2}-\left|\frac{y}{3z^{2}}\right|\left(% \frac{u}{\tfrac{2}{3}-u}\right)^{1/2}=\frac{y^{2}}{6wz^{4}}-2w^{3}-2w^{2}}}
8u^{3}-4u^{2}-\left|\frac{y}{3z^{2}}\right|\left(\frac{u}{\tfrac{2}{3}-u}\right)^{1/2} = \frac{y^{2}}{6wz^{4}}-2w^{3}-2w^{2}		 

8*(u)^(3)- 4*(u)^(2)-abs((y)/(3*(x + y*I)^(2)))*((u)/((2)/(3)- u))^(1/2) = ((y)^(2))/(6*w*(x + y*I)^(4))- 2*(u -(2)/(3)+(((2)/(3)- u)^(2)+abs((y)/(6*(x + y*I)^(2)))*(((2)/(3)- u)/(u))^(1/2))^(1/2))^(3)- 2*(u -(2)/(3)+(((2)/(3)- u)^(2)+abs((y)/(6*(x + y*I)^(2)))*(((2)/(3)- u)/(u))^(1/2))^(1/2))^(2)
 
8*(u)^(3)- 4*(u)^(2)-Abs[Divide[y,3*(x + y*I)^(2)]]*(Divide[u,Divide[2,3]- u])^(1/2) == Divide[(y)^(2),6*w*(x + y*I)^(4)]- 2*(u -Divide[2,3]+((Divide[2,3]- u)^(2)+Abs[Divide[y,6*(x + y*I)^(2)]]*(Divide[Divide[2,3]- u,u])^(1/2))^(1/2))^(3)- 2*(u -Divide[2,3]+((Divide[2,3]- u)^(2)+Abs[Divide[y,6*(x + y*I)^(2)]]*(Divide[Divide[2,3]- u,u])^(1/2))^(1/2))^(2)
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E14 0 < u 0 𝑒 {\displaystyle{\displaystyle 0<u}}
0 < u		 

0 < u
 
0 < u
 Skipped - no semantic math Skipped - no semantic math - -
36.5#Ex11 Y ⁒ ( u , X ) = 8 ⁒ u - 24 ⁒ u 2 + X ⁒ u - 1 6 ( u ⁒ ( u - 1 3 ) ) 1 / 2 π‘Œ 𝑒 𝑋 8 𝑒 24 superscript 𝑒 2 𝑋 𝑒 1 6 superscript 𝑒 𝑒 1 3 1 2 {\displaystyle{\displaystyle Y(u,X)=8u-24u^{2}+X\dfrac{u-\tfrac{1}{6}}{\left(u% \left(u-\tfrac{1}{3}\right)\right)^{1/2}}}}
Y(u,X) = 8u-24u^{2}+X\dfrac{u-\tfrac{1}{6}}{\left(u\left(u-\tfrac{1}{3}\right)\right)^{1/2}}		 

Y(u ,((x - y)/(x + y*I)^(2))) = 8*u - 24*(u)^(2)+((x - y)/(x + y*I)^(2))*(u -(1)/(6))/((u*(u -(1)/(3)))^(1/2))
 
Y[u ,((x - y)/(x + y*I)^(2))] == 8*u - 24*(u)^(2)+((x - y)/(x + y*I)^(2))*Divide[u -Divide[1,6],(u*(u -Divide[1,3]))^(1/2)]
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E17 Y S ⁒ ( X ) = Y ⁒ ( u , | X | ) subscript π‘Œ S 𝑋 π‘Œ 𝑒 𝑋 {\displaystyle{\displaystyle Y_{\mathrm{S}}(X)=Y(u,|X|)}}
Y_{\mathrm{S}}(X) = Y(u,|X|)		 

Y[S](((x - y)/(x + y*I)^(2))) = Y(u ,abs((x - y)/(x + y*I)^(2)))
 
Subscript[Y, S][((x - y)/(x + y*I)^(2))] == Y[u ,Abs[(x - y)/(x + y*I)^(2)]]
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E19 Y S ⁒ ( X ) = Y ⁒ ( - u , - | X | ) subscript π‘Œ S 𝑋 π‘Œ 𝑒 𝑋 {\displaystyle{\displaystyle Y_{\mathrm{S}}(X)=Y(-u,-|X|)}}
Y_{\mathrm{S}}(X) = Y(-u,-|X|)		 

Y[S](((x - y)/(x + y*I)^(2))) = Y(- u , -abs((x - y)/(x + y*I)^(2)))
 
Subscript[Y, S][((x - y)/(x + y*I)^(2))] == Y[- u , -Abs[(x - y)/(x + y*I)^(2)]]
 Skipped - no semantic math Skipped - no semantic math - -
36.5.E21 w = ( 1 3 + u ) ⁒ ( 1 - ( 1 - | X | 12 ⁒ u 1 / 2 ⁒ ( 1 3 + u ) 3 / 2 ) 1 / 2 ) 𝑀 1 3 𝑒 1 superscript 1 𝑋 12 superscript 𝑒 1 2 superscript 1 3 𝑒 3 2 1 2 {\displaystyle{\displaystyle w=(\tfrac{1}{3}+u)\left(1-\left(1-\dfrac{|X|}{12u% ^{1/2}(\tfrac{1}{3}+u)^{3/2}}\right)^{1/2}\right)}}
w = (\tfrac{1}{3}+u)\left(1-\left(1-\dfrac{|X|}{12u^{1/2}(\tfrac{1}{3}+u)^{3/2}}\right)^{1/2}\right)		 

(u -(2)/(3)+(((2)/(3)- u)^(2)+abs((y)/(6*(x + y*I)^(2)))*(((2)/(3)- u)/(u))^(1/2))^(1/2)) = ((1)/(3)+ u)*(1 -(1 -(abs((x - y)/(x + y*I)^(2)))/(12*(u)^(1/2)*((1)/(3)+ u)^(3/2)))^(1/2))
 
(u -Divide[2,3]+((Divide[2,3]- u)^(2)+Abs[Divide[y,6*(x + y*I)^(2)]]*(Divide[Divide[2,3]- u,u])^(1/2))^(1/2)) == (Divide[1,3]+ u)*(1 -(1 -Divide[Abs[(x - y)/(x + y*I)^(2)],12*(u)^(1/2)*(Divide[1,3]+ u)^(3/2)])^(1/2))
 Skipped - no semantic math Skipped - no semantic math - -
Retrieved from "https://lct.wmflabs.org/w/index.php?title=36.5&oldid=58565"