Elliptic Integrals - 19.22 Quadratic Transformations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.22.E1 R F ⁡ ( 0 , x 2 , y 2 ) = R F ⁡ ( 0 , x ⁢ y , a 2 ) Carlson-integral-RF 0 superscript 𝑥 2 superscript 𝑦 2 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 {\displaystyle{\displaystyle R_{F}\left(0,x^{2},y^{2}\right)=R_{F}\left(0,xy,a% ^{2}\right)}}
\CarlsonsymellintRF@{0}{x^{2}}{y^{2}} = \CarlsonsymellintRF@{0}{xy}{a^{2}}		 

0.5*int(1/(sqrt(t+0)*sqrt(t+(x)^(2))*sqrt(t+(y)^(2))), t = 0..infinity) = 0.5*int(1/(sqrt(t+0)*sqrt(t+x*y)*sqrt(t+(a)^(2))), t = 0..infinity)

EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]/Sqrt[(y)^(2)-0] == EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]
Aborted Failure Skipped - Because timed out
Failed [102 / 108]
Result: Complex[0.1731783664325578, 0.8740191847640398]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[0.4406854652170371, 0.9732684211375591]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, -0.5]}

... skip entries to safe data
19.22.E2 2 ⁢ R G ⁡ ( 0 , x 2 , y 2 ) = 4 ⁢ R G ⁡ ( 0 , x ⁢ y , a 2 ) - x ⁢ y ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) 2 Carlson-integral-RG 0 superscript 𝑥 2 superscript 𝑦 2 4 Carlson-integral-RG 0 𝑥 𝑦 superscript 𝑎 2 𝑥 𝑦 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 {\displaystyle{\displaystyle 2R_{G}\left(0,x^{2},y^{2}\right)=4R_{G}\left(0,xy% ,a^{2}\right)-xyR_{F}\left(0,xy,a^{2}\right)}}
2\CarlsonsymellintRG@{0}{x^{2}}{y^{2}} = 4\CarlsonsymellintRG@{0}{xy}{a^{2}}-xy\CarlsonsymellintRF@{0}{xy}{a^{2}}		 

Error

2*Sqrt[(y)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(y)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]+Cot[ArcCos[Sqrt[0/(y)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(y)^(2)]]]^2]) == 4*Sqrt[(a)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(a)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]+Cot[ArcCos[Sqrt[0/(a)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(a)^(2)]]]^2])- x*y*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]
Missing Macro Error Failure -
Failed [108 / 108]
Result: Complex[-0.848574889541176, -1.6278775384876862]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, -1.5]}


Result: -2.356194490192345
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.22.E3 2 ⁢ y 2 ⁢ R D ⁡ ( 0 , x 2 , y 2 ) = 1 4 ⁢ ( y 2 - x 2 ) ⁢ R D ⁡ ( 0 , x ⁢ y , a 2 ) + 3 ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) 2 superscript 𝑦 2 Carlson-integral-RD 0 superscript 𝑥 2 superscript 𝑦 2 1 4 superscript 𝑦 2 superscript 𝑥 2 Carlson-integral-RD 0 𝑥 𝑦 superscript 𝑎 2 3 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 {\displaystyle{\displaystyle 2y^{2}R_{D}\left(0,x^{2},y^{2}\right)=\tfrac{1}{4% }(y^{2}-x^{2})R_{D}\left(0,xy,a^{2}\right)+3R_{F}\left(0,xy,a^{2}\right)}}
2y^{2}\CarlsonsymellintRD@{0}{x^{2}}{y^{2}} = \tfrac{1}{4}(y^{2}-x^{2})\CarlsonsymellintRD@{0}{xy}{a^{2}}+3\CarlsonsymellintRF@{0}{xy}{a^{2}}		 

Error

2*(y)^(2)* 3*(EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)])/(((y)^(2)-(x)^(2))*((y)^(2)-0)^(1/2)) == Divide[1,4]*((y)^(2)- (x)^(2))*3*(EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]-EllipticE[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)])/(((a)^(2)-x*y)*((a)^(2)-0)^(1/2))+ 3*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]
Missing Macro Error Failure -
Failed [108 / 108]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, -1.5]}


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

... skip entries to safe data
19.22.E4 ( p + 2 - p - 2 ) ⁢ R J ⁡ ( 0 , x 2 , y 2 , p 2 ) = 2 ⁢ ( p + 2 - a 2 ) ⁢ R J ⁡ ( 0 , x ⁢ y , a 2 , p + 2 ) - 3 ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) + 3 ⁢ π / ( 2 ⁢ p ) superscript subscript 𝑝 2 superscript subscript 𝑝 2 Carlson-integral-RJ 0 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑝 2 2 superscript subscript 𝑝 2 superscript 𝑎 2 Carlson-integral-RJ 0 𝑥 𝑦 superscript 𝑎 2 superscript subscript 𝑝 2 3 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 3 𝜋 2 𝑝 {\displaystyle{\displaystyle(p_{+}^{2}-p_{-}^{2})R_{J}\left(0,x^{2},y^{2},p^{2% }\right)=2(p_{+}^{2}-a^{2})R_{J}\left(0,xy,a^{2},p_{+}^{2}\right)-3R_{F}\left(% 0,xy,a^{2}\right)+3\pi/(2p)}}
(p_{+}^{2}-p_{-}^{2})\CarlsonsymellintRJ@{0}{x^{2}}{y^{2}}{p^{2}} = 2(p_{+}^{2}-a^{2})\CarlsonsymellintRJ@{0}{xy}{a^{2}}{p_{+}^{2}}-3\CarlsonsymellintRF@{0}{xy}{a^{2}}+3\pi/(2p)		 

Error

((Subscript[p, +])^(2)- (Subscript[p, -])^(2))*3*((y)^(2)-0)/((y)^(2)-(p)^(2))*(EllipticPi[((y)^(2)-(p)^(2))/((y)^(2)-0),ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)])/Sqrt[(y)^(2)-0] == 2*((Subscript[p, +])^(2)- (a)^(2))*3*((a)^(2)-0)/((a)^(2)-(Subscript[p, +])^(2))*(EllipticPi[((a)^(2)-(Subscript[p, +])^(2))/((a)^(2)-0),ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)])/Sqrt[(a)^(2)-0]- 3*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]+ 3*Pi/(2*p)
Missing Macro Error Failure - Error
19.22.E4 ( p - 2 - p + 2 ) ⁢ R J ⁡ ( 0 , x 2 , y 2 , p 2 ) = 2 ⁢ ( p - 2 - a 2 ) ⁢ R J ⁡ ( 0 , x ⁢ y , a 2 , p - 2 ) - 3 ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) + 3 ⁢ π / ( 2 ⁢ p ) superscript subscript 𝑝 2 superscript subscript 𝑝 2 Carlson-integral-RJ 0 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑝 2 2 superscript subscript 𝑝 2 superscript 𝑎 2 Carlson-integral-RJ 0 𝑥 𝑦 superscript 𝑎 2 superscript subscript 𝑝 2 3 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 3 𝜋 2 𝑝 {\displaystyle{\displaystyle(p_{-}^{2}-p_{+}^{2})R_{J}\left(0,x^{2},y^{2},p^{2% }\right)=2(p_{-}^{2}-a^{2})R_{J}\left(0,xy,a^{2},p_{-}^{2}\right)-3R_{F}\left(% 0,xy,a^{2}\right)+3\pi/(2p)}}
(p_{-}^{2}-p_{+}^{2})\CarlsonsymellintRJ@{0}{x^{2}}{y^{2}}{p^{2}} = 2(p_{-}^{2}-a^{2})\CarlsonsymellintRJ@{0}{xy}{a^{2}}{p_{-}^{2}}-3\CarlsonsymellintRF@{0}{xy}{a^{2}}+3\pi/(2p)		 

Error

((Subscript[p, -])^(2)- (Subscript[p, +])^(2))*3*((y)^(2)-0)/((y)^(2)-(p)^(2))*(EllipticPi[((y)^(2)-(p)^(2))/((y)^(2)-0),ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)])/Sqrt[(y)^(2)-0] == 2*((Subscript[p, -])^(2)- (a)^(2))*3*((a)^(2)-0)/((a)^(2)-(Subscript[p, -])^(2))*(EllipticPi[((a)^(2)-(Subscript[p, -])^(2))/((a)^(2)-0),ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)])/Sqrt[(a)^(2)-0]- 3*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]+ 3*Pi/(2*p)
Missing Macro Error Failure - Error
19.22#Ex1 p + ⁢ p - = p ⁢ a subscript 𝑝 subscript 𝑝 𝑝 𝑎 {\displaystyle{\displaystyle p_{+}p_{-}=pa}}
p_{+}p_{-} = pa		 

p[+]*p[-] = p*a
 
Subscript[p, +]*Subscript[p, -] == p*a
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex2 p + 2 + p - 2 = p 2 + x ⁢ y superscript subscript 𝑝 2 superscript subscript 𝑝 2 superscript 𝑝 2 𝑥 𝑦 {\displaystyle{\displaystyle p_{+}^{2}+p_{-}^{2}=p^{2}+xy}}
p_{+}^{2}+p_{-}^{2} = p^{2}+xy		 

(p[+])^(2)+ (p[-])^(2) = (p)^(2)+ x*y
 
(Subscript[p, +])^(2)+ (Subscript[p, -])^(2) == (p)^(2)+ x*y
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex3 p + 2 - p - 2 = ( p 2 - x 2 ) ⁢ ( p 2 - y 2 ) superscript subscript 𝑝 2 superscript subscript 𝑝 2 superscript 𝑝 2 superscript 𝑥 2 superscript 𝑝 2 superscript 𝑦 2 {\displaystyle{\displaystyle p_{+}^{2}-p_{-}^{2}=\sqrt{(p^{2}-x^{2})(p^{2}-y^{% 2})}}}
p_{+}^{2}-p_{-}^{2} = \sqrt{(p^{2}-x^{2})(p^{2}-y^{2})}		 

(p[+])^(2)- (p[-])^(2) = sqrt(((p)^(2)- (x)^(2))*((p)^(2)- (y)^(2)))
 
(Subscript[p, +])^(2)- (Subscript[p, -])^(2) == Sqrt[((p)^(2)- (x)^(2))*((p)^(2)- (y)^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex4 4 ⁢ ( p + 2 - a 2 ) = ( p 2 - x 2 + p 2 - y 2 ) 2 4 superscript subscript 𝑝 2 superscript 𝑎 2 superscript superscript 𝑝 2 superscript 𝑥 2 superscript 𝑝 2 superscript 𝑦 2 2 {\displaystyle{\displaystyle 4(p_{+}^{2}-a^{2})=(\sqrt{p^{2}-x^{2}}+\sqrt{p^{2% }-y^{2}})^{2}}}
4(p_{+}^{2}-a^{2}) = (\sqrt{p^{2}-x^{2}}+\sqrt{p^{2}-y^{2}})^{2}		 

4*((p[+])^(2)- (a)^(2)) = (sqrt((p)^(2)- (x)^(2))+sqrt((p)^(2)- (y)^(2)))^(2)
 
4*((Subscript[p, +])^(2)- (a)^(2)) == (Sqrt[(p)^(2)- (x)^(2)]+Sqrt[(p)^(2)- (y)^(2)])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.22.E7 2 ⁢ p 2 ⁢ R J ⁡ ( 0 , x 2 , y 2 , p 2 ) = v + ⁢ v - ⁢ R J ⁡ ( 0 , x ⁢ y , a 2 , v + 2 ) + 3 ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) 2 superscript 𝑝 2 Carlson-integral-RJ 0 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑝 2 subscript 𝑣 subscript 𝑣 Carlson-integral-RJ 0 𝑥 𝑦 superscript 𝑎 2 subscript superscript 𝑣 2 3 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 {\displaystyle{\displaystyle 2p^{2}R_{J}\left(0,x^{2},y^{2},p^{2}\right)=v_{+}% v_{-}R_{J}\left(0,xy,a^{2},v^{2}_{+}\right)+3R_{F}\left(0,xy,a^{2}\right)}}
2p^{2}\CarlsonsymellintRJ@{0}{x^{2}}{y^{2}}{p^{2}} = v_{+}v_{-}\CarlsonsymellintRJ@{0}{xy}{a^{2}}{v^{2}_{+}}+3\CarlsonsymellintRF@{0}{xy}{a^{2}}		 v + = ( p 2 + x ⁢ y ) / ( 2 ⁢ p ) , v - = ( p 2 - x ⁢ y ) / ( 2 ⁢ p ) formulae-sequence subscript 𝑣 superscript 𝑝 2 𝑥 𝑦 2 𝑝 subscript 𝑣 superscript 𝑝 2 𝑥 𝑦 2 𝑝 {\displaystyle{\displaystyle v_{+}=(p^{2}+xy)/(2p),v_{-}=(p^{2}-xy)/(2p)}} 
Error

2*(p)^(2)* 3*((y)^(2)-0)/((y)^(2)-(p)^(2))*(EllipticPi[((y)^(2)-(p)^(2))/((y)^(2)-0),ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(y)^(2)]],((y)^(2)-(x)^(2))/((y)^(2)-0)])/Sqrt[(y)^(2)-0] == Subscript[v, +]*Subscript[v, -]*3*((a)^(2)-0)/((a)^(2)-(Subscript[v, +])^(2))*(EllipticPi[((a)^(2)-(Subscript[v, +])^(2))/((a)^(2)-0),ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]-EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)])/Sqrt[(a)^(2)-0]+ 3*EllipticF[ArcCos[Sqrt[0/(a)^(2)]],((a)^(2)-x*y)/((a)^(2)-0)]/Sqrt[(a)^(2)-0]
Missing Macro Error Failure - Error
19.22.E8 2 π ⁢ R F ⁡ ( 0 , a 0 2 , g 0 2 ) = 1 M ⁡ ( a 0 , g 0 ) 2 𝜋 Carlson-integral-RF 0 superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 {\displaystyle{\displaystyle\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}% \right)=\frac{1}{M\left(a_{0},g_{0}\right)}}}
\frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}} = \frac{1}{\AGM@{a_{0}}{g_{0}}}		 

(2)/(Pi)*0.5*int(1/(sqrt(t+0)*sqrt(t+(a[0])^(2))*sqrt(t+(g[0])^(2))), t = 0..infinity) = (1)/(GaussAGM(a[0], g[0]))

Error
Aborted Missing Macro Error Skipped - Because timed out -
19.22.E9 1 M ⁡ ( a 0 , g 0 ) ⁢ ( a 0 2 - ∑ n = 0 ∞ 2 n - 1 ⁢ c n 2 ) = 1 M ⁡ ( a 0 , g 0 ) ⁢ ( a 1 2 - ∑ n = 2 ∞ 2 n - 1 ⁢ c n 2 ) 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 superscript subscript 𝑎 0 2 superscript subscript 𝑛 0 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 superscript subscript 𝑎 1 2 superscript subscript 𝑛 2 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 {\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{0}^{2}% -\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{M\left(a_{0},g_{0}\right)% }\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}
\frac{1}{\AGM@{a_{0}}{g_{0}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right) = \frac{1}{\AGM@{a_{0}}{g_{0}}}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)		 

(1)/(GaussAGM(a[0], g[0]))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity)) = (1)/(GaussAGM(a[0], g[0]))*((a[1])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 2..infinity))

Error
Failure Missing Macro Error Error -
19.22#Ex5 Q 0 = 1 subscript 𝑄 0 1 {\displaystyle{\displaystyle Q_{0}=1}}
Q_{0} = 1		 

Q[0] = 1
 
Subscript[Q, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex6 Q n + 1 = 1 2 ⁢ Q n ⁢ a n - g n a n + g n subscript 𝑄 𝑛 1 1 2 subscript 𝑄 𝑛 subscript 𝑎 𝑛 subscript 𝑔 𝑛 subscript 𝑎 𝑛 subscript 𝑔 𝑛 {\displaystyle{\displaystyle Q_{n+1}=\tfrac{1}{2}Q_{n}\frac{a_{n}-g_{n}}{a_{n}% +g_{n}}}}
Q_{n+1} = \tfrac{1}{2}Q_{n}\frac{a_{n}-g_{n}}{a_{n}+g_{n}}		 

Q[n + 1] = (1)/(2)*Q[n]*(a[n]- g[n])/(a[n]+ g[n])
 
Subscript[Q, n + 1] == Divide[1,2]*Subscript[Q, n]*Divide[Subscript[a, n]- Subscript[g, n],Subscript[a, n]+ Subscript[g, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex7 p n + 1 = p n 2 + a n ⁢ g n 2 ⁢ p n subscript 𝑝 𝑛 1 superscript subscript 𝑝 𝑛 2 subscript 𝑎 𝑛 subscript 𝑔 𝑛 2 subscript 𝑝 𝑛 {\displaystyle{\displaystyle p_{n+1}=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}}}
p_{n+1} = \frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}		 

p[n + 1] = ((p[n])^(2)+ a[n]*g[n])/(2*p[n])
 
Subscript[p, n + 1] == Divide[(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n],2*Subscript[p, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex8 ε n = p n 2 - a n ⁢ g n p n 2 + a n ⁢ g n subscript 𝜀 𝑛 superscript subscript 𝑝 𝑛 2 subscript 𝑎 𝑛 subscript 𝑔 𝑛 superscript subscript 𝑝 𝑛 2 subscript 𝑎 𝑛 subscript 𝑔 𝑛 {\displaystyle{\displaystyle\varepsilon_{n}=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^% {2}+a_{n}g_{n}}}}
\varepsilon_{n} = \frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}		 

varepsilon[n] = ((p[n])^(2)- a[n]*g[n])/((p[n])^(2)+ a[n]*g[n])
 
Subscript[\[CurlyEpsilon], n] == Divide[(Subscript[p, n])^(2)- Subscript[a, n]*Subscript[g, n],(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex9 Q 0 = 1 subscript 𝑄 0 1 {\displaystyle{\displaystyle Q_{0}=1}}
Q_{0} = 1		 

Q[0] = 1
 
Subscript[Q, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex10 Q n + 1 = 1 2 ⁢ Q n ⁢ ε n subscript 𝑄 𝑛 1 1 2 subscript 𝑄 𝑛 subscript 𝜀 𝑛 {\displaystyle{\displaystyle Q_{n+1}=\tfrac{1}{2}Q_{n}\varepsilon_{n}}}
Q_{n+1} = \tfrac{1}{2}Q_{n}\varepsilon_{n}		 

Q[n + 1] = (1)/(2)*Q[n]*varepsilon[n]
 
Subscript[Q, n + 1] == Divide[1,2]*Subscript[Q, n]*Subscript[\[CurlyEpsilon], n]
 Skipped - no semantic math Skipped - no semantic math - -
19.22.E15 p 0 2 = a 0 2 ⁢ ( q 0 2 + g 0 2 ) / ( q 0 2 + a 0 2 ) superscript subscript 𝑝 0 2 superscript subscript 𝑎 0 2 superscript subscript 𝑞 0 2 superscript subscript 𝑔 0 2 superscript subscript 𝑞 0 2 superscript subscript 𝑎 0 2 {\displaystyle{\displaystyle p_{0}^{2}=a_{0}^{2}(q_{0}^{2}+g_{0}^{2})/(q_{0}^{% 2}+a_{0}^{2})}}
p_{0}^{2} = a_{0}^{2}(q_{0}^{2}+g_{0}^{2})/(q_{0}^{2}+a_{0}^{2})		 

(p[0])^(2) = (a[0])^(2)*((q[0])^(2)+ (g[0])^(2))/((q[0])^(2)+ (a[0])^(2))
 
(Subscript[p, 0])^(2) == (Subscript[a, 0])^(2)*((Subscript[q, 0])^(2)+ (Subscript[g, 0])^(2))/((Subscript[q, 0])^(2)+ (Subscript[a, 0])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex11 a = ( x + y ) / 2 𝑎 𝑥 𝑦 2 {\displaystyle{\displaystyle a=(x+y)/2}}
a = (x+y)/2		 

a = (x + y)/2
 
a == (x + y)/2
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex12 2 ⁢ z + = ( z + x ) ⁢ ( z + y ) + ( z - x ) ⁢ ( z - y ) 2 subscript 𝑧 𝑧 𝑥 𝑧 𝑦 𝑧 𝑥 𝑧 𝑦 {\displaystyle{\displaystyle 2z_{+}=\sqrt{(z+x)(z+y)}+\sqrt{(z-x)(z-y)}}}
2z_{+} = \sqrt{(z+x)(z+y)}+\sqrt{(z-x)(z-y)}		 

2*x + y*I[+] = sqrt(((x + y*I)+ x)*((x + y*I)+ y))+sqrt(((x + y*I)- x)*((x + y*I)- y))
 
2*Subscript[x + y*I, +] == Sqrt[((x + y*I)+ x)*((x + y*I)+ y)]+Sqrt[((x + y*I)- x)*((x + y*I)- y)]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex13 z + ⁢ z - = z ⁢ a subscript 𝑧 subscript 𝑧 𝑧 𝑎 {\displaystyle{\displaystyle z_{+}z_{-}=za}}
z_{+}z_{-} = za		 

z[+]*z[-] = z*a
 
Subscript[z, +]*Subscript[z, -] == z*a
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex14 z + 2 + z - 2 = z 2 + x ⁢ y superscript subscript 𝑧 2 superscript subscript 𝑧 2 superscript 𝑧 2 𝑥 𝑦 {\displaystyle{\displaystyle z_{+}^{2}+z_{-}^{2}=z^{2}+xy}}
z_{+}^{2}+z_{-}^{2} = z^{2}+xy		 

(x + y*I[+])^(2)+(x + y*I[-])^(2) = (x + y*I)^(2)+ x*y
 
(Subscript[x + y*I, +])^(2)+(Subscript[x + y*I, -])^(2) == (x + y*I)^(2)+ x*y
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex15 z + 2 - z - 2 = ( z 2 - x 2 ) ⁢ ( z 2 - y 2 ) superscript subscript 𝑧 2 superscript subscript 𝑧 2 superscript 𝑧 2 superscript 𝑥 2 superscript 𝑧 2 superscript 𝑦 2 {\displaystyle{\displaystyle z_{+}^{2}-z_{-}^{2}=\sqrt{(z^{2}-x^{2})(z^{2}-y^{% 2})}}}
z_{+}^{2}-z_{-}^{2} = \sqrt{(z^{2}-x^{2})(z^{2}-y^{2})}		 

(x + y*I[+])^(2)-(x + y*I[-])^(2) = sqrt(((x + y*I)^(2)- (x)^(2))*((x + y*I)^(2)- (y)^(2)))
 
(Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2) == Sqrt[((x + y*I)^(2)- (x)^(2))*((x + y*I)^(2)- (y)^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex16 4 ⁢ ( z + 2 - a 2 ) = ( z 2 - x 2 + z 2 - y 2 ) 2 4 superscript subscript 𝑧 2 superscript 𝑎 2 superscript superscript 𝑧 2 superscript 𝑥 2 superscript 𝑧 2 superscript 𝑦 2 2 {\displaystyle{\displaystyle 4(z_{+}^{2}-a^{2})=(\sqrt{z^{2}-x^{2}}+\sqrt{z^{2% }-y^{2}})^{2}}}
4(z_{+}^{2}-a^{2}) = (\sqrt{z^{2}-x^{2}}+\sqrt{z^{2}-y^{2}})^{2}		 

4*((x + y*I[+])^(2)- (a)^(2)) = (sqrt((x + y*I)^(2)- (x)^(2))+sqrt((x + y*I)^(2)- (y)^(2)))^(2)
 
4*((Subscript[x + y*I, +])^(2)- (a)^(2)) == (Sqrt[(x + y*I)^(2)- (x)^(2)]+Sqrt[(x + y*I)^(2)- (y)^(2)])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.22.E18 R F ⁡ ( x 2 , y 2 , z 2 ) = R F ⁡ ( a 2 , z - 2 , z + 2 ) Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 Carlson-integral-RF superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 {\displaystyle{\displaystyle R_{F}\left(x^{2},y^{2},z^{2}\right)=R_{F}\left(a^% {2},z_{-}^{2},z_{+}^{2}\right)}}
\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}} = \CarlsonsymellintRF@{a^{2}}{z_{-}^{2}}{z_{+}^{2}}		 

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

EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)] == EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, +])^(2)]],((Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2))/((Subscript[x + y*I, +])^(2)-(a)^(2))]/Sqrt[(Subscript[x + y*I, +])^(2)-(a)^(2)]
Error Failure - Error
19.22.E19 ( z + 2 - z - 2 ) ⁢ R D ⁡ ( x 2 , y 2 , z 2 ) = 2 ⁢ ( z + 2 - a 2 ) ⁢ R D ⁡ ( a 2 , z - 2 , z + 2 ) - 3 ⁢ R F ⁡ ( x 2 , y 2 , z 2 ) + ( 3 / z ) superscript subscript 𝑧 2 superscript subscript 𝑧 2 Carlson-integral-RD superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 2 superscript subscript 𝑧 2 superscript 𝑎 2 Carlson-integral-RD superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 3 Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 3 𝑧 {\displaystyle{\displaystyle(z_{+}^{2}-z_{-}^{2})R_{D}\left(x^{2},y^{2},z^{2}% \right)={2(z_{+}^{2}-a^{2})}R_{D}\left(a^{2},z_{-}^{2},z_{+}^{2}\right)-3R_{F}% \left(x^{2},y^{2},z^{2}\right)+(3/z)}}
(z_{+}^{2}-z_{-}^{2})\CarlsonsymellintRD@{x^{2}}{y^{2}}{z^{2}} = {2(z_{+}^{2}-a^{2})}\CarlsonsymellintRD@{a^{2}}{z_{-}^{2}}{z_{+}^{2}}-3\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}}+(3/z)		 

Error

((Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2))*3*(EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]-EllipticE[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))])/(((x + y*I)^(2)-(y)^(2))*((x + y*I)^(2)-(x)^(2))^(1/2)) == 2*((Subscript[x + y*I, +])^(2)- (a)^(2))*3*(EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, +])^(2)]],((Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2))/((Subscript[x + y*I, +])^(2)-(a)^(2))]-EllipticE[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, +])^(2)]],((Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2))/((Subscript[x + y*I, +])^(2)-(a)^(2))])/(((Subscript[x + y*I, +])^(2)-(Subscript[x + y*I, -])^(2))*((Subscript[x + y*I, +])^(2)-(a)^(2))^(1/2))- 3*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)]+(3/(x + y*I))
Missing Macro Error Failure - Error
19.22.E19 ( z - 2 - z + 2 ) ⁢ R D ⁡ ( x 2 , y 2 , z 2 ) = 2 ⁢ ( z - 2 - a 2 ) ⁢ R D ⁡ ( a 2 , z + 2 , z - 2 ) - 3 ⁢ R F ⁡ ( x 2 , y 2 , z 2 ) + ( 3 / z ) superscript subscript 𝑧 2 superscript subscript 𝑧 2 Carlson-integral-RD superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 2 superscript subscript 𝑧 2 superscript 𝑎 2 Carlson-integral-RD superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 3 Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 3 𝑧 {\displaystyle{\displaystyle(z_{-}^{2}-z_{+}^{2})R_{D}\left(x^{2},y^{2},z^{2}% \right)={2(z_{-}^{2}-a^{2})}R_{D}\left(a^{2},z_{+}^{2},z_{-}^{2}\right)-3R_{F}% \left(x^{2},y^{2},z^{2}\right)+(3/z)}}
(z_{-}^{2}-z_{+}^{2})\CarlsonsymellintRD@{x^{2}}{y^{2}}{z^{2}} = {2(z_{-}^{2}-a^{2})}\CarlsonsymellintRD@{a^{2}}{z_{+}^{2}}{z_{-}^{2}}-3\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}}+(3/z)		 

Error

((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))*3*(EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]-EllipticE[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))])/(((x + y*I)^(2)-(y)^(2))*((x + y*I)^(2)-(x)^(2))^(1/2)) == 2*((Subscript[x + y*I, -])^(2)- (a)^(2))*3*(EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))]-EllipticE[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))])/(((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))*((Subscript[x + y*I, -])^(2)-(a)^(2))^(1/2))- 3*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)]+(3/(x + y*I))
Missing Macro Error Failure - Error
19.22.E20 ( p + 2 - p - 2 ) ⁢ R J ⁡ ( x 2 , y 2 , z 2 , p 2 ) = 2 ⁢ ( p + 2 - a 2 ) ⁢ R J ⁡ ( a 2 , z + 2 , z - 2 , p + 2 ) - 3 ⁢ R F ⁡ ( x 2 , y 2 , z 2 ) + 3 ⁢ R C ⁡ ( z 2 , p 2 ) superscript subscript 𝑝 2 superscript subscript 𝑝 2 Carlson-integral-RJ superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 superscript 𝑝 2 2 superscript subscript 𝑝 2 superscript 𝑎 2 Carlson-integral-RJ superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 superscript subscript 𝑝 2 3 Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 3 Carlson-integral-RC superscript 𝑧 2 superscript 𝑝 2 {\displaystyle{\displaystyle(p_{+}^{2}-p_{-}^{2})R_{J}\left(x^{2},y^{2},z^{2},% p^{2}\right)=2(p_{+}^{2}-a^{2})R_{J}\left(a^{2},z_{+}^{2},z_{-}^{2},p_{+}^{2}% \right)-3R_{F}\left(x^{2},y^{2},z^{2}\right)+3R_{C}\left(z^{2},p^{2}\right)}}
(p_{+}^{2}-p_{-}^{2})\CarlsonsymellintRJ@{x^{2}}{y^{2}}{z^{2}}{p^{2}} = 2(p_{+}^{2}-a^{2})\CarlsonsymellintRJ@{a^{2}}{z_{+}^{2}}{z_{-}^{2}}{p_{+}^{2}}-3\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}}+3\CarlsonellintRC@{z^{2}}{p^{2}}		 

Error

((Subscript[p, +])^(2)- (Subscript[p, -])^(2))*3*((x + y*I)^(2)-(x)^(2))/((x + y*I)^(2)-(p)^(2))*(EllipticPi[((x + y*I)^(2)-(p)^(2))/((x + y*I)^(2)-(x)^(2)),ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]-EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))])/Sqrt[(x + y*I)^(2)-(x)^(2)] == 2*((Subscript[p, +])^(2)- (a)^(2))*3*((Subscript[x + y*I, -])^(2)-(a)^(2))/((Subscript[x + y*I, -])^(2)-(Subscript[p, +])^(2))*(EllipticPi[((Subscript[x + y*I, -])^(2)-(Subscript[p, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2)),ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))]-EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))])/Sqrt[(Subscript[x + y*I, -])^(2)-(a)^(2)]- 3*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)]+ 3*1/Sqrt[(p)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x + y*I)^(2))/((p)^(2))]
Missing Macro Error Failure - Error
19.22.E20 ( p - 2 - p + 2 ) ⁢ R J ⁡ ( x 2 , y 2 , z 2 , p 2 ) = 2 ⁢ ( p - 2 - a 2 ) ⁢ R J ⁡ ( a 2 , z + 2 , z - 2 , p - 2 ) - 3 ⁢ R F ⁡ ( x 2 , y 2 , z 2 ) + 3 ⁢ R C ⁡ ( z 2 , p 2 ) superscript subscript 𝑝 2 superscript subscript 𝑝 2 Carlson-integral-RJ superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 superscript 𝑝 2 2 superscript subscript 𝑝 2 superscript 𝑎 2 Carlson-integral-RJ superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 superscript subscript 𝑝 2 3 Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 3 Carlson-integral-RC superscript 𝑧 2 superscript 𝑝 2 {\displaystyle{\displaystyle(p_{-}^{2}-p_{+}^{2})R_{J}\left(x^{2},y^{2},z^{2},% p^{2}\right)=2(p_{-}^{2}-a^{2})R_{J}\left(a^{2},z_{+}^{2},z_{-}^{2},p_{-}^{2}% \right)-3R_{F}\left(x^{2},y^{2},z^{2}\right)+3R_{C}\left(z^{2},p^{2}\right)}}
(p_{-}^{2}-p_{+}^{2})\CarlsonsymellintRJ@{x^{2}}{y^{2}}{z^{2}}{p^{2}} = 2(p_{-}^{2}-a^{2})\CarlsonsymellintRJ@{a^{2}}{z_{+}^{2}}{z_{-}^{2}}{p_{-}^{2}}-3\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}}+3\CarlsonellintRC@{z^{2}}{p^{2}}		 

Error

((Subscript[p, -])^(2)- (Subscript[p, +])^(2))*3*((x + y*I)^(2)-(x)^(2))/((x + y*I)^(2)-(p)^(2))*(EllipticPi[((x + y*I)^(2)-(p)^(2))/((x + y*I)^(2)-(x)^(2)),ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]-EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))])/Sqrt[(x + y*I)^(2)-(x)^(2)] == 2*((Subscript[p, -])^(2)- (a)^(2))*3*((Subscript[x + y*I, -])^(2)-(a)^(2))/((Subscript[x + y*I, -])^(2)-(Subscript[p, -])^(2))*(EllipticPi[((Subscript[x + y*I, -])^(2)-(Subscript[p, -])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2)),ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))]-EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))])/Sqrt[(Subscript[x + y*I, -])^(2)-(a)^(2)]- 3*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)]+ 3*1/Sqrt[(p)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x + y*I)^(2))/((p)^(2))]
Missing Macro Error Failure - Error
19.22.E21 2 ⁢ R G ⁡ ( x 2 , y 2 , z 2 ) = 4 ⁢ R G ⁡ ( a 2 , z + 2 , z - 2 ) - x ⁢ y ⁢ R F ⁡ ( x 2 , y 2 , z 2 ) - z 2 Carlson-integral-RG superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 4 Carlson-integral-RG superscript 𝑎 2 superscript subscript 𝑧 2 superscript subscript 𝑧 2 𝑥 𝑦 Carlson-integral-RF superscript 𝑥 2 superscript 𝑦 2 superscript 𝑧 2 𝑧 {\displaystyle{\displaystyle 2R_{G}\left(x^{2},y^{2},z^{2}\right)=4R_{G}\left(% a^{2},z_{+}^{2},z_{-}^{2}\right)-xyR_{F}\left(x^{2},y^{2},z^{2}\right)-z}}
2\CarlsonsymellintRG@{x^{2}}{y^{2}}{z^{2}} = 4\CarlsonsymellintRG@{a^{2}}{z_{+}^{2}}{z_{-}^{2}}-xy\CarlsonsymellintRF@{x^{2}}{y^{2}}{z^{2}}-z		 

Error

2*Sqrt[(x + y*I)^(2)-(x)^(2)]*(EllipticE[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]+(Cot[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]]])^2*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]+Cot[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]]]^2]) == 4*Sqrt[(Subscript[x + y*I, -])^(2)-(a)^(2)]*(EllipticE[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))]+(Cot[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]]])^2*EllipticF[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]],((Subscript[x + y*I, -])^(2)-(Subscript[x + y*I, +])^(2))/((Subscript[x + y*I, -])^(2)-(a)^(2))]+Cot[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(a)^(2)/(Subscript[x + y*I, -])^(2)]]]^2])- x*y*EllipticF[ArcCos[Sqrt[(x)^(2)/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-(x)^(2))]/Sqrt[(x + y*I)^(2)-(x)^(2)]-(x + y*I)
Missing Macro Error Failure - Error
19.22.E22 R C ⁡ ( x 2 , y 2 ) = R C ⁡ ( a 2 , a ⁢ y ) Carlson-integral-RC superscript 𝑥 2 superscript 𝑦 2 Carlson-integral-RC superscript 𝑎 2 𝑎 𝑦 {\displaystyle{\displaystyle R_{C}\left(x^{2},y^{2}\right)=R_{C}\left(a^{2},ay% \right)}}
\CarlsonellintRC@{x^{2}}{y^{2}} = \CarlsonellintRC@{a^{2}}{ay}		 

Error

1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))] == 1/Sqrt[a*y]*Hypergeometric2F1[1/2,1/2,3/2,1-((a)^(2))/(a*y)]
Missing Macro Error Failure -
Failed [108 / 108]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[y, -1.5]}


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

... skip entries to safe data
19.22#Ex17 x + y = 2 ⁢ a 𝑥 𝑦 2 𝑎 {\displaystyle{\displaystyle x+y=2a}}
x+y = 2a		 

x + y = 2*a
 
x + y == 2*a
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex18 x - y = ( 2 / a ) ⁢ ( a 2 - z + 2 ) ⁢ ( a 2 - z - 2 ) 𝑥 𝑦 2 𝑎 superscript 𝑎 2 superscript subscript 𝑧 2 superscript 𝑎 2 superscript subscript 𝑧 2 {\displaystyle{\displaystyle x-y=(\ifrac{2}{a})\sqrt{(a^{2}-z_{+}^{2})(a^{2}-z% _{-}^{2})}}}
x-y = (\ifrac{2}{a})\sqrt{(a^{2}-z_{+}^{2})(a^{2}-z_{-}^{2})}		 

x - y = ((2)/(a))*sqrt(((a)^(2)-(x + y*I[+])^(2))*((a)^(2)-(x + y*I[-])^(2)))
 
x - y == (Divide[2,a])*Sqrt[((a)^(2)-(Subscript[x + y*I, +])^(2))*((a)^(2)-(Subscript[x + y*I, -])^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.22#Ex19 z = z + ⁢ z - / a 𝑧 subscript 𝑧 subscript 𝑧 𝑎 {\displaystyle{\displaystyle z=\ifrac{z_{+}z_{-}}{a}}}
z = \ifrac{z_{+}z_{-}}{a}		 

z = (z[+]*z[-])/(a)
 
z == Divide[Subscript[z, +]*Subscript[z, -],a]
 Skipped - no semantic math Skipped - no semantic math - -
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