Elliptic Integrals - 19.32 Conformal Map onto a Rectangle

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.32.E1 z ⁒ ( p ) = R F ⁑ ( p - x 1 , p - x 2 , p - x 3 ) 𝑧 𝑝 Carlson-integral-RF 𝑝 subscript π‘₯ 1 𝑝 subscript π‘₯ 2 𝑝 subscript π‘₯ 3 {\displaystyle{\displaystyle z(p)=R_{F}\left(p-x_{1},p-x_{2},p-x_{3}\right)}}
z(p) = \CarlsonsymellintRF@{p-x_{1}}{p-x_{2}}{p-x_{3}}		 

(x + y*I)*(p) = 0.5*int(1/(sqrt(t+p - x[1])*sqrt(t+p - x[2])*sqrt(t+p - x[3])), t = 0..infinity)

(x + y*I)*(p) == EllipticF[ArcCos[Sqrt[p - Subscript[x, 1]/p - Subscript[x, 3]]],(p - Subscript[x, 3]-p - Subscript[x, 2])/(p - Subscript[x, 3]-p - Subscript[x, 1])]/Sqrt[p - Subscript[x, 3]-p - Subscript[x, 1]]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[-0.7208699572238464, -0.7193085577979393]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.3758216901446034, -2.446030868401005]
Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.32.E3 x 1 > x 2 subscript π‘₯ 1 subscript π‘₯ 2 {\displaystyle{\displaystyle x_{1}>x_{2}}}
x_{1} > x_{2}		 

x[1] > x[2]
 
Subscript[x, 1] > Subscript[x, 2]
 Skipped - no semantic math Skipped - no semantic math - -
19.32#Ex1 z ⁒ ( ∞ ) = 0 𝑧 0 {\displaystyle{\displaystyle z(\infty)=0}}
z(\infty) = 0		 

z(infinity) = 0
 
z[Infinity] == 0
 Skipped - no semantic math Skipped - no semantic math - -
19.32#Ex3 z ⁒ ( x 2 ) = z ⁒ ( x 1 ) + z ⁒ ( x 3 ) 𝑧 subscript π‘₯ 2 𝑧 subscript π‘₯ 1 𝑧 subscript π‘₯ 3 {\displaystyle{\displaystyle z(x_{2})=z(x_{1})+z(x_{3})}}
z(x_{2}) = z(x_{1})+z(x_{3})		 

(x + y*I)*(x[2]) = (x + y*I)*(x[1])+(x + y*I)*(x[3])
 
(x + y*I)*(Subscript[x, 2]) == (x + y*I)*(Subscript[x, 1])+(x + y*I)*(Subscript[x, 3])
 Skipped - no semantic math Skipped - no semantic math - -
19.32#Ex4 z ⁒ ( x 3 ) = R F ⁑ ( x 3 - x 1 , x 3 - x 2 , 0 ) 𝑧 subscript π‘₯ 3 Carlson-integral-RF subscript π‘₯ 3 subscript π‘₯ 1 subscript π‘₯ 3 subscript π‘₯ 2 0 {\displaystyle{\displaystyle z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0% \right)}}
z(x_{3}) = \CarlsonsymellintRF@{x_{3}-x_{1}}{x_{3}-x_{2}}{0}		 

(x + y*I)*(x[3]) = 0.5*int(1/(sqrt(t+x[3]- x[1])*sqrt(t+x[3]- x[2])*sqrt(t+0)), t = 0..infinity)

(x + y*I)*(Subscript[x, 3]) == EllipticF[ArcCos[Sqrt[Subscript[x, 3]- Subscript[x, 1]/0]],(0-Subscript[x, 3]- Subscript[x, 2])/(0-Subscript[x, 3]- Subscript[x, 1])]/Sqrt[0-Subscript[x, 3]- Subscript[x, 1]]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[1.024519052838329, -0.27451905283832906], Times[Complex[-0.25881904510252074, -0.9659258262890683], EllipticF[DirectedInfinity[], 1.0]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[0.27451905283832917, 1.0245190528383288], Times[Complex[-0.7239434227163943, -0.9434614369855119], EllipticF[DirectedInfinity[], 1.0]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.32#Ex4 R F ⁑ ( x 3 - x 1 , x 3 - x 2 , 0 ) = - i ⁒ R F ⁑ ( 0 , x 1 - x 3 , x 2 - x 3 ) Carlson-integral-RF subscript π‘₯ 3 subscript π‘₯ 1 subscript π‘₯ 3 subscript π‘₯ 2 0 𝑖 Carlson-integral-RF 0 subscript π‘₯ 1 subscript π‘₯ 3 subscript π‘₯ 2 subscript π‘₯ 3 {\displaystyle{\displaystyle R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{% F}\left(0,x_{1}-x_{3},x_{2}-x_{3}\right)}}
\CarlsonsymellintRF@{x_{3}-x_{1}}{x_{3}-x_{2}}{0} = -i\CarlsonsymellintRF@{0}{x_{1}-x_{3}}{x_{2}-x_{3}}		 

0.5*int(1/(sqrt(t+x[3]- x[1])*sqrt(t+x[3]- x[2])*sqrt(t+0)), t = 0..infinity) = - I*0.5*int(1/(sqrt(t+0)*sqrt(t+x[1]- x[3])*sqrt(t+x[2]- x[3])), t = 0..infinity)

EllipticF[ArcCos[Sqrt[Subscript[x, 3]- Subscript[x, 1]/0]],(0-Subscript[x, 3]- Subscript[x, 2])/(0-Subscript[x, 3]- Subscript[x, 1])]/Sqrt[0-Subscript[x, 3]- Subscript[x, 1]] == - I*EllipticF[ArcCos[Sqrt[0/Subscript[x, 2]- Subscript[x, 3]]],(Subscript[x, 2]- Subscript[x, 3]-Subscript[x, 1]- Subscript[x, 3])/(Subscript[x, 2]- Subscript[x, 3]-0)]/Sqrt[Subscript[x, 2]- Subscript[x, 3]-0]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[-0.4754994064110389, 1.6461555153586378], Times[Complex[0.7239434227163943, 0.9434614369855119], EllipticF[DirectedInfinity[], 1.0]]]
Test Values: {Rule[Subscript[x, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[x, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=19.32&oldid=58386"