Elliptic Integrals - 19.14 Reduction of General Elliptic Integrals

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.14.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{1}^{x}\frac{\diff{t}}{\sqrt{t^{3}-1}} = 3^{-1/4}\incellintFk@{\phi}{k}}
\int_{1}^{x}\frac{\diff{t}}{\sqrt{t^{3}-1}} = 3^{-1/4}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}, k^{2} = \dfrac{2-\sqrt{3}}{4}} 
int((1)/(sqrt((t)^(3)- 1)), t = 1..x) = (3)^(- 1/4)* EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[(t)^(3)- 1]], {t, 1, x}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]
Failure Aborted Error Skipped - Because timed out
19.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{1}\frac{\diff{t}}{\sqrt{1-t^{3}}} = 3^{-1/4}\incellintFk@{\phi}{k}}
\int_{x}^{1}\frac{\diff{t}}{\sqrt{1-t^{3}}} = 3^{-1/4}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}, k^{2} = \dfrac{2+\sqrt{3}}{4}} 
int((1)/(sqrt(1 - (t)^(3))), t = x..1) = (3)^(- 1/4)* EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[1 - (t)^(3)]], {t, x, 1}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]
Failure Aborted Error Skip - No test values generated
19.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\frac{\diff{t}}{\sqrt{1+t^{4}}} = \frac{\sign@{x}}{2}\incellintFk@{\phi}{k}}
\int_{0}^{x}\frac{\diff{t}}{\sqrt{1+t^{4}}} = \frac{\sign@{x}}{2}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{1-x^{2}}{1+x^{2}}, k^{2} = \dfrac{1}{2}} 
int((1)/(sqrt(1 + (t)^(4))), t = 0..x) = (signum(x))/(2)*EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[1 + (t)^(4)]], {t, 0, x}, GenerateConditions->None] == Divide[Sign[x],2]*EllipticF[\[Phi], (k)^2]
Failure Failure Error Skip - No test values generated
19.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{y}^{x}\frac{\diff{t}}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}} = \frac{1}{\sqrt{\gamma-\alpha}}\incellintFk@{\phi}{k}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}} = \frac{1}{\sqrt{\gamma-\alpha}}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = \ifrac{(\gamma-\beta)}{(\gamma-\alpha)}} 
int((1)/(sqrt((a[1]+ b[1]*(t)^(2))*(a[2]+ b[2]*(t)^(2)))), t = y..x) = (1)/(sqrt(gamma - alpha))*EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(t)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(t)^(2))]], {t, y, x}, GenerateConditions->None] == Divide[1,Sqrt[\[Gamma]- \[Alpha]]]*EllipticF[\[Phi], (k)^2]
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
19.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{U^{2}+\gamma}}
\sin^{2}@@{\phi} = \frac{\gamma-\alpha}{U^{2}+\gamma}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = (gamma - alpha)/((U)^(2)+ gamma)

(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],(U)^(2)+ \[Gamma]]
Failure Failure
Failed [300 / 300]
Result: 1.144207228+.1616580578*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I}


Result: .2329549284-1.570148532*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.0397570908067482, -1.0061601508735134]
Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.7911458419033055, -1.4391726141222814]
Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (x^{2}-y^{2})U = x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}}
(x^{2}-y^{2})U = x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((x)^(2)- (y)^(2))*U = x*sqrt((a[1]+ b[1]*(y)^(2))*(a[2]+ b[2]*(y)^(2)))+ y*sqrt((a[1]+ b[1]*(x)^(2))*(a[2]+ b[2]*(x)^(2)))
 
((x)^(2)- (y)^(2))*U == x*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(y)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]+ y*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(x)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*(x)^(2))/(a[1]*a[2]+ gamma*(x)^(2))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*(x)^(2),Subscript[a, 1]*Subscript[a, 2]+ \[Gamma]*(x)^(2)]
Failure Failure
Failed [300 / 300]
Result: 1.560947444+.1288116535*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = 1/2*3^(1/2)+1/2*I}


Result: 2.678639127-1.794319469*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.3471528039744003, -1.172411794219179]
Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.5030688086095803, -1.7852795940180226]
Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}}
\sin^{2}@@{\phi} = \frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = (gamma - alpha)/(b[1]*b[2]*(y)^(2)+ gamma)

(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],Subscript[b, 1]*Subscript[b, 2]*(y)^(2)+ \[Gamma]]
Failure Failure
Failed [300 / 300]
Result: .8585159693+.3113806358*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = 1/2*3^(1/2)+1/2*I}


Result: .2216600130+.2500138214*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.683185473382228, -0.7175596041712626]
Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.5335538340604822, -1.7418837307419275]
Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*((x)^(2)- (y)^(2)))/(gamma*((x)^(2)- (y)^(2))- a[1]*(a[2]+ b[2]*(x)^(2)))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((x)^(2)- (y)^(2)),\[Gamma]*((x)^(2)- (y)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]
Failure Failure Manual Skip! Skipped - Because timed out
19.14.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*((y)^(2)- (x)^(2)))/(gamma*((y)^(2)- (x)^(2))- a[1]*(a[2]+ b[2]*(y)^(2)))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((y)^(2)- (x)^(2)),\[Gamma]*((y)^(2)- (x)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]
Failure Failure Manual Skip! Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=19.14&oldid=58371"