Orthogonal Polynomials - 19.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica
19.2.E2 r ⁢ ( s , t ) = ( p 1 + p 2 ⁢ s ) ⁢ ( p 3 - p 4 ⁢ s ) ⁢ s ( p 3 + p 4 ⁢ s ) ⁢ ( p 3 - p 4 ⁢ s ) ⁢ s 𝑟 𝑠 𝑡 subscript 𝑝 1 subscript 𝑝 2 𝑠 subscript 𝑝 3 subscript 𝑝 4 𝑠 𝑠 subscript 𝑝 3 subscript 𝑝 4 𝑠 subscript 𝑝 3 subscript 𝑝 4 𝑠 𝑠 {\displaystyle{\displaystyle r(s,t)=\frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3% }+p_{4}s)(p_{3}-p_{4}s)s}}}
r(s,t) = \frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}		 

r(s , t) = ((p[1]+ p[2]*s)*(p[3]- p[4]*s)*s)/((p[3]+ p[4]*s)*(p[3]- p[4]*s)*s)
 
r[s , t] == Divide[(Subscript[p, 1]+ Subscript[p, 2]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s,(Subscript[p, 3]+ Subscript[p, 4]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s]
 Skipped - no semantic math Skipped - no semantic math - -
19.2.E4 F ⁡ ( ϕ , k ) = ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 {\displaystyle{\displaystyle F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\mathrm% {d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}
\incellintFk@{\phi}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}		 

EllipticF(sin(phi), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)

EllipticF[\[Phi], (k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Failure Aborted
Failed [6 / 30]
Result: Float(infinity)
Test Values: {phi = -2, k = 1}


Result: .2e-9-.5175477340*I
Test Values: {phi = -2, k = 2}

... skip entries to safe data
 Skipped - Because timed out
19.2.E4 ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ = ∫ 0 sin ⁡ ϕ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 superscript subscript 0 italic-ϕ 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 {\displaystyle{\displaystyle\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{% 2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt% {1-k^{2}t^{2}}}}}
\int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}		 

int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))

Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
Failure Aborted
Failed [6 / 30]
Result: Float(-infinity)
Test Values: {phi = -2, k = 1}


Result: -.2e-9+.5175477340*I
Test Values: {phi = -2, k = 2}

... skip entries to safe data
 Skipped - Because timed out
19.2.E5 E ⁡ ( ϕ , k ) = ∫ 0 ϕ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ d θ elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript subscript 0 italic-ϕ 1 superscript 𝑘 2 2 𝜃 𝜃 {\displaystyle{\displaystyle E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}% {\sin^{2}}\theta}\mathrm{d}\theta\\ }}
\incellintEk@{\phi}{k} = \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\		 

EllipticE(sin(phi), k) = int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)

EllipticE[\[Phi], (k)^2] == Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
19.2.E5 ∫ 0 ϕ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ d θ = ∫ 0 sin ⁡ ϕ 1 - k 2 ⁢ t 2 1 - t 2 ⁢ d t superscript subscript 0 italic-ϕ 1 superscript 𝑘 2 2 𝜃 𝜃 superscript subscript 0 italic-ϕ 1 superscript 𝑘 2 superscript 𝑡 2 1 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}% \mathrm{d}\theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\mathrm{d}t}}
\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\ = \int_{0}^{\sin@@{\phi}}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\diff{t}		 

int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi) = int((sqrt(1 - (k)^(2)* (t)^(2)))/(sqrt(1 - (t)^(2))), t = 0..sin(phi))

Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[Sqrt[1 - (k)^(2)* (t)^(2)],Sqrt[1 - (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
19.2.E6 D ⁡ ( ϕ , k ) = ∫ 0 ϕ sin 2 ⁡ θ ⁢ d θ 1 - k 2 ⁢ sin 2 ⁡ θ elliptic-integral-third-kind-D italic-ϕ 𝑘 superscript subscript 0 italic-ϕ 2 𝜃 𝜃 1 superscript 𝑘 2 2 𝜃 {\displaystyle{\displaystyle D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{% 2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}
\incellintDk@{\phi}{k} = \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}		 

(EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2 = int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)

Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4] == Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
19.2.E6 ∫ 0 ϕ sin 2 ⁡ θ ⁢ d θ 1 - k 2 ⁢ sin 2 ⁡ θ = ∫ 0 sin ⁡ ϕ t 2 ⁢ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 superscript subscript 0 italic-ϕ 2 𝜃 𝜃 1 superscript 𝑘 2 2 𝜃 superscript subscript 0 italic-ϕ superscript 𝑡 2 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 {\displaystyle{\displaystyle\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}% \theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{% d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}}
\int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}		 

int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))

Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
19.2.E6 ∫ 0 sin ⁡ ϕ t 2 ⁢ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 = ( F ⁡ ( ϕ , k ) - E ⁡ ( ϕ , k ) ) / k 2 superscript subscript 0 italic-ϕ superscript 𝑡 2 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{\sqrt{1% -t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))/k^{2}}}
\int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} = (\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})/k^{2}		 

int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi)) = (EllipticF(sin(phi), k)- EllipticE(sin(phi), k))/(k)^(2)

Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None] == (EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])/(k)^(2)
Failure Aborted Successful [Tested: 0] Skipped - Because timed out
19.2.E7 Π ⁡ ( ϕ , α 2 , k ) = ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ ( 1 - α 2 ⁢ sin 2 ⁡ θ ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 1 superscript 𝛼 2 2 𝜃 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}% \frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}% \theta)}}}
\incellintPik@{\phi}{\alpha^{2}}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})}		 

EllipticPi(sin(phi), (alpha)^(2), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi)

EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.2.E7 ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ ( 1 - α 2 ⁢ sin 2 ⁡ θ ) = ∫ 0 sin ⁡ ϕ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 ⁢ ( 1 - α 2 ⁢ t 2 ) superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 1 superscript 𝛼 2 2 𝜃 superscript subscript 0 italic-ϕ 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 1 superscript 𝛼 2 superscript 𝑡 2 {\displaystyle{\displaystyle\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{% 2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}\theta)}=\int_{0}^{\sin\phi}\frac{% \mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}}}
\int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}		 

int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))*(1 - (alpha)^(2)* (t)^(2))), t = 0..sin(phi))

Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]*(1 - \[Alpha]^(2)* (t)^(2))], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.2#Ex1 K ⁡ ( k ) = F ⁡ ( π / 2 , k ) complete-elliptic-integral-first-kind-K 𝑘 elliptic-integral-first-kind-F 𝜋 2 𝑘 {\displaystyle{\displaystyle K\left(k\right)=F\left(\pi/2,k\right)}}
\compellintKk@{k} = \incellintFk@{\pi/2}{k}		 

EllipticK(k) = EllipticF(sin(Pi/2), k)

EllipticK[(k)^2] == EllipticF[Pi/2, (k)^2]
Successful Successful - Successful [Tested: 3]
19.2#Ex2 E ⁡ ( k ) = E ⁡ ( π / 2 , k ) complete-elliptic-integral-second-kind-E 𝑘 elliptic-integral-second-kind-E 𝜋 2 𝑘 {\displaystyle{\displaystyle E\left(k\right)=E\left(\pi/2,k\right)}}
\compellintEk@{k} = \incellintEk@{\pi/2}{k}		 

EllipticE(k) = EllipticE(sin(Pi/2), k)

EllipticE[(k)^2] == EllipticE[Pi/2, (k)^2]
Successful Successful - Successful [Tested: 3]
19.2#Ex3 D ⁡ ( k ) = D ⁡ ( π / 2 , k ) complete-elliptic-integral-D 𝑘 elliptic-integral-third-kind-D 𝜋 2 𝑘 {\displaystyle{\displaystyle D\left(k\right)=D\left(\pi/2,k\right)}}
\compellintDk@{k} = \incellintDk@{\pi/2}{k}		 

(EllipticK(k) - EllipticE(k))/(k)^2 = (EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2

Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4]
Successful Successful - Successful [Tested: 3]
19.2#Ex3 D ⁡ ( π / 2 , k ) = ( K ⁡ ( k ) - E ⁡ ( k ) ) / k 2 elliptic-integral-third-kind-D 𝜋 2 𝑘 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle D\left(\pi/2,k\right)=(K\left(k\right)-E\left(k% \right))/k^{2}}}
\incellintDk@{\pi/2}{k} = (\compellintKk@{k}-\compellintEk@{k})/k^{2}		 

(EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2 = (EllipticK(k)- EllipticE(k))/(k)^(2)

Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4] == (EllipticK[(k)^2]- EllipticE[(k)^2])/(k)^(2)
Successful Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-0.08185805455243832, 0.4541460103381725]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.2#Ex4 Π ⁡ ( α 2 , k ) = Π ⁡ ( π / 2 , α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 elliptic-integral-third-kind-Pi 𝜋 2 superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\Pi\left(\pi/2,\alpha% ^{2},k\right)}}
\compellintPik@{\alpha^{2}}{k} = \incellintPik@{\pi/2}{\alpha^{2}}{k}		 

EllipticPi((alpha)^(2), k) = EllipticPi(sin(Pi/2), (alpha)^(2), k)

EllipticPi[\[Alpha]^(2), (k)^2] == EllipticPi[\[Alpha]^(2), Pi/2,(k)^2]
Successful Successful - Successful [Tested: 9]
19.2#Ex5 K ′ ⁡ ( k ) = K ⁡ ( k ′ ) complementary-complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle{K^{\prime}}\left(k\right)=K\left(k^{\prime}\right% )}}
\ccompellintKk@{k} = \compellintKk@{k^{\prime}}		 

EllipticCK(k) = EllipticK(sqrt(1 - (k)^(2)))

EllipticK[1-(k)^2] == EllipticK[(Sqrt[1 - (k)^(2)])^2]
Successful Successful - Successful [Tested: 3]
19.2#Ex6 E ′ ⁡ ( k ) = E ⁡ ( k ′ ) complementary-complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-second-kind-E superscript 𝑘 ′ {\displaystyle{\displaystyle{E^{\prime}}\left(k\right)=E\left(k^{\prime}\right% )}}
\ccompellintEk@{k} = \compellintEk@{k^{\prime}}		 

EllipticCE(k) = EllipticE(sqrt(1 - (k)^(2)))

EllipticE[1-(k)^2] == EllipticE[(Sqrt[1 - (k)^(2)])^2]
Successful Successful - Successful [Tested: 3]
19.2#Ex7 k ′ = 1 - k 2 superscript 𝑘 ′ 1 superscript 𝑘 2 {\displaystyle{\displaystyle k^{\prime}=\sqrt{1-k^{2}}}}
k^{\prime} = \sqrt{1-k^{2}}		 

sqrt(1 - (k)^(2)) = sqrt(1 - (k)^(2))

Sqrt[1 - (k)^(2)] == Sqrt[1 - (k)^(2)]
Successful Successful - Successful [Tested: 3]
19.2#Ex8 F ⁡ ( m ⁢ π + ϕ , k ) = 2 ⁢ m ⁢ K ⁡ ( k ) + F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-first-kind-K 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(m\pi+\phi,k\right)=2mK\left(k\right)+F% \left(\phi,k\right)}}
\incellintFk@{m\pi+\phi}{k} = 2m\compellintKk@{k}+\incellintFk@{\phi}{k}		 

EllipticF(sin(m*Pi + phi), k) = 2*m*EllipticK(k)+ EllipticF(sin(phi), k)

EllipticF[m*Pi + \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]+ EllipticF[\[Phi], (k)^2]
Failure Failure Error
Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.2#Ex8 F ⁡ ( m ⁢ π - ϕ , k ) = 2 ⁢ m ⁢ K ⁡ ( k ) - F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-first-kind-K 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(m\pi-\phi,k\right)=2mK\left(k\right)-F% \left(\phi,k\right)}}
\incellintFk@{m\pi-\phi}{k} = 2m\compellintKk@{k}-\incellintFk@{\phi}{k}		 

EllipticF(sin(m*Pi - phi), k) = 2*m*EllipticK(k)- EllipticF(sin(phi), k)

EllipticF[m*Pi - \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]- EllipticF[\[Phi], (k)^2]
Failure Failure Error
Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.2#Ex9 E ⁡ ( m ⁢ π + ϕ , k ) = 2 ⁢ m ⁢ E ⁡ ( k ) + E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-second-kind-E 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(m\pi+\phi,k\right)=2mE\left(k\right)+E% \left(\phi,k\right)}}
\incellintEk@{m\pi+\phi}{k} = 2m\compellintEk@{k}+\incellintEk@{\phi}{k}		 

EllipticE(sin(m*Pi + phi), k) = 2*m*EllipticE(k)+ EllipticE(sin(phi), k)

EllipticE[m*Pi + \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]+ EllipticE[\[Phi], (k)^2]
Failure Failure
Failed [90 / 90]
Result: -3.717960670-.6751929261*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1}


Result: -4.000000000-.3e-9*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2}

... skip entries to safe data
 Successful [Tested: 90]
19.2#Ex9 E ⁡ ( m ⁢ π - ϕ , k ) = 2 ⁢ m ⁢ E ⁡ ( k ) - E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-second-kind-E 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(m\pi-\phi,k\right)=2mE\left(k\right)-E% \left(\phi,k\right)}}
\incellintEk@{m\pi-\phi}{k} = 2m\compellintEk@{k}-\incellintEk@{\phi}{k}		 

EllipticE(sin(m*Pi - phi), k) = 2*m*EllipticE(k)- EllipticE(sin(phi), k)

EllipticE[m*Pi - \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]- EllipticE[\[Phi], (k)^2]
Failure Failure
Failed [90 / 90]
Result: -.2820393315+.6751929264*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1}


Result: -4.000000000-.4e-9*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2}

... skip entries to safe data
 Successful [Tested: 90]
19.2#Ex10 D ⁡ ( m ⁢ π + ϕ , k ) = 2 ⁢ m ⁢ D ⁡ ( k ) + D ⁡ ( ϕ , k ) elliptic-integral-third-kind-D 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-D 𝑘 elliptic-integral-third-kind-D italic-ϕ 𝑘 {\displaystyle{\displaystyle D\left(m\pi+\phi,k\right)=2mD\left(k\right)+D% \left(\phi,k\right)}}
\incellintDk@{m\pi+\phi}{k} = 2m\compellintDk@{k}+\incellintDk@{\phi}{k}		 

(EllipticF(sin(m*Pi + phi), k) - EllipticE(sin(m*Pi + phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 + (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2

Divide[EllipticF[m*Pi + \[Phi], (k)^2] - EllipticE[m*Pi + \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]+ Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]
Failure Failure Error
Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.2#Ex10 D ⁡ ( m ⁢ π - ϕ , k ) = 2 ⁢ m ⁢ D ⁡ ( k ) - D ⁡ ( ϕ , k ) elliptic-integral-third-kind-D 𝑚 𝜋 italic-ϕ 𝑘 2 𝑚 complete-elliptic-integral-D 𝑘 elliptic-integral-third-kind-D italic-ϕ 𝑘 {\displaystyle{\displaystyle D\left(m\pi-\phi,k\right)=2mD\left(k\right)-D% \left(\phi,k\right)}}
\incellintDk@{m\pi-\phi}{k} = 2m\compellintDk@{k}-\incellintDk@{\phi}{k}		 

(EllipticF(sin(m*Pi - phi), k) - EllipticE(sin(m*Pi - phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 - (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2

Divide[EllipticF[m*Pi - \[Phi], (k)^2] - EllipticE[m*Pi - \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]- Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]
Failure Failure Error
Failed [30 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.2.E16 ∫ 0 arctan ⁡ x d θ ( cos 2 ⁡ θ + p ⁢ sin 2 ⁡ θ ) ⁢ cos 2 ⁡ θ + k c 2 ⁢ sin 2 ⁡ θ = Π ⁡ ( arctan ⁡ x , 1 - p , k ) superscript subscript 0 𝑥 𝜃 2 𝜃 𝑝 2 𝜃 2 𝜃 superscript subscript 𝑘 𝑐 2 2 𝜃 elliptic-integral-third-kind-Pi 𝑥 1 𝑝 𝑘 {\displaystyle{\displaystyle\int_{0}^{\operatorname{arctan}x}\frac{\mathrm{d}% \theta}{({\cos^{2}}\theta+p{\sin^{2}}\theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{% \sin^{2}}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right)}}
\int_{0}^{\atan@@{x}}\frac{\diff{\theta}}{(\cos^{2}@@{\theta}+p\sin^{2}@@{\theta})\sqrt{\cos^{2}@@{\theta}+k_{c}^{2}\sin^{2}@@{\theta}}} = \incellintPik@{\atan@@{x}}{1-p}{k}		 x 2 ≠ - 1 / p superscript 𝑥 2 1 𝑝 {\displaystyle{\displaystyle x^{2}\neq-1/p}} 
int((1)/(((cos(theta))^(2)+ p*(sin(theta))^(2))*sqrt((cos(theta))^(2)+ (k[c])^(2)*(sin(theta))^(2))), theta = 0..arctan(x)) = EllipticPi(sin(arctan(x)), 1 - p, k)

Integrate[Divide[1,((Cos[\[Theta]])^(2)+ p*(Sin[\[Theta]])^(2))*Sqrt[(Cos[\[Theta]])^(2)+ (Subscript[k, c])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, ArcTan[x]}, GenerateConditions->None] == EllipticPi[1 - p, ArcTan[x],(k)^2]
Error Aborted - Skipped - Because timed out
19.2.E17 R C ⁡ ( x , y ) = 1 2 ⁢ ∫ 0 ∞ d t t + x ⁢ ( t + y ) Carlson-integral-RC 𝑥 𝑦 1 2 superscript subscript 0 𝑡 𝑡 𝑥 𝑡 𝑦 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty% }\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)}}}
\CarlsonellintRC@{x}{y} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t+x}(t+y)}		 

Error

1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,2]*Integrate[Divide[1,Sqrt[t + x]*(t + y)], {t, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [12 / 18]
Result: Complex[-1.0177225554447185, 0.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.2.E18 R C ⁡ ( x , y ) = 1 y - x ⁢ arctan ⁡ y - x x Carlson-integral-RC 𝑥 𝑦 1 𝑦 𝑥 𝑦 𝑥 𝑥 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}% \operatorname{arctan}\sqrt{\frac{y-x}{x}}}}
\CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}}		 0 ≤ x , x < y formulae-sequence 0 𝑥 𝑥 𝑦 {\displaystyle{\displaystyle 0\leq x,x<y}} 
Error

1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]]
Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 3]
19.2.E18 1 y - x ⁢ arctan ⁡ y - x x = 1 y - x ⁢ arccos ⁡ x / y 1 𝑦 𝑥 𝑦 𝑥 𝑥 1 𝑦 𝑥 𝑥 𝑦 {\displaystyle{\displaystyle\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{% \frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y}}}
\frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}} = \frac{1}{\sqrt{y-x}}\acos@@{\sqrt{x/y}}		 0 ≤ x , x < y formulae-sequence 0 𝑥 𝑥 𝑦 {\displaystyle{\displaystyle 0\leq x,x<y}} 
(1)/(sqrt(y - x))*arctan(sqrt((y - x)/(x))) = (1)/(sqrt(y - x))*arccos(sqrt(x/y))
 
Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]] == Divide[1,Sqrt[y - x]]*ArcCos[Sqrt[x/y]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
19.2.E19 R C ⁡ ( x , y ) = 1 x - y ⁢ arctanh ⁡ x - y x Carlson-integral-RC 𝑥 𝑦 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑦 𝑥 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}% \operatorname{arctanh}\sqrt{\frac{x-y}{x}}}}
\CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}}		 0 < y , y < x formulae-sequence 0 𝑦 𝑦 𝑥 {\displaystyle{\displaystyle 0<y,y<x}} 
Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]]
 Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 3]
19.2.E19 1 x - y ⁢ arctanh ⁡ x - y x = 1 x - y ⁢ ln ⁡ x + x - y y 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑦 𝑥 1 𝑥 𝑦 𝑥 𝑥 𝑦 𝑦 {\displaystyle{\displaystyle\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{% \frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}}}
\frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}}		 0 < y , y < x formulae-sequence 0 𝑦 𝑦 𝑥 {\displaystyle{\displaystyle 0<y,y<x}} 
(1)/(sqrt(x - y))*arctanh(sqrt((x - y)/(x))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(y)))
 
Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[y]]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
19.2.E20 R C ⁡ ( x , y ) = x x - y ⁢ R C ⁡ ( x - y , - y ) Carlson-integral-RC 𝑥 𝑦 𝑥 𝑥 𝑦 Carlson-integral-RC 𝑥 𝑦 𝑦 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\sqrt{\frac{x}{x-y}}R_{C}% \left(x-y,-y\right)}}
\CarlsonellintRC@{x}{y} = \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y}		 y < 0 , 0 ≤ x formulae-sequence 𝑦 0 0 𝑥 {\displaystyle{\displaystyle y<0,0\leq x}} 
Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)]
 Missing Macro Error Failure Skip - symbolical successful subtest
Failed [9 / 9]
Result: Complex[-1.0177225554447187, -0.906899682117109]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Complex[-1.862459718905424, -1.1107207345395915]
Test Values: {Rule[x, 1.5], Rule[y, -0.5]}

... skip entries to safe data
19.2.E20 x x - y ⁢ R C ⁡ ( x - y , - y ) = 1 x - y ⁢ arctanh ⁡ x x - y 𝑥 𝑥 𝑦 Carlson-integral-RC 𝑥 𝑦 𝑦 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑥 𝑦 {\displaystyle{\displaystyle\sqrt{\frac{x}{x-y}}R_{C}\left(x-y,-y\right)=\frac% {1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x-y}}}}
\sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}}		 y < 0 , 0 ≤ x formulae-sequence 𝑦 0 0 𝑥 {\displaystyle{\displaystyle y<0,0\leq x}} 
Error
 
Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]]
 Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 9]
19.2.E20 1 x - y ⁢ arctanh ⁡ x x - y = 1 x - y ⁢ ln ⁡ x + x - y - y 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑥 𝑦 1 𝑥 𝑦 𝑥 𝑥 𝑦 𝑦 {\displaystyle{\displaystyle\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{% \frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}}}
\frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}}		 y < 0 , 0 ≤ x formulae-sequence 𝑦 0 0 𝑥 {\displaystyle{\displaystyle y<0,0\leq x}} 
(1)/(sqrt(x - y))*arctanh(sqrt((x)/(x - y))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(- y)))
 
Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[- y]]]
 Failure Failure Successful [Tested: 9] Successful [Tested: 9]
19.2.E21 R C ⁡ ( x , y ) = ∫ 0 1 ( v 2 ⁢ x + ( 1 - v 2 ) ⁢ y ) - 1 / 2 ⁢ d v Carlson-integral-RC 𝑥 𝑦 superscript subscript 0 1 superscript superscript 𝑣 2 𝑥 1 superscript 𝑣 2 𝑦 1 2 𝑣 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\int_{0}^{1}(v^{2}x+(1-v^{2% })y)^{-1/2}\mathrm{d}v}}
\CarlsonellintRC@{x}{y} = \int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\diff{v}		 

Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Integrate[((v)^(2)* x +(1 - (v)^(2))*y)^(- 1/2), {v, 0, 1}, GenerateConditions->None]
 Missing Macro Error Aborted - Skipped - Because timed out
19.2.E22 R C ⁡ ( x , y ) = 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( y , x ⁢ cos 2 ⁡ θ + y ⁢ sin 2 ⁡ θ ) ⁢ d θ Carlson-integral-RC 𝑥 𝑦 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑦 𝑥 2 𝜃 𝑦 2 𝜃 𝜃 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{2}{\pi}\int_{0}^{\pi/% 2}R_{C}\left(y,x{\cos^{2}}\theta+y{\sin^{2}}\theta\right)\mathrm{d}\theta}}
\CarlsonellintRC@{x}{y} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{x\cos^{2}@@{\theta}+y\sin^{2}@@{\theta}}\diff{\theta}		 

Error
 
1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[2,Pi]*Integrate[1/Sqrt[x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y)/(x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None]
 Missing Macro Error Aborted - Skipped - Because timed out
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