Elliptic Integrals - 19.24 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.24.E1 ln ⁡ 4 ≤ z ⁢ R F ⁡ ( 0 , y , z ) + ln ⁡ y / z 4 𝑧 Carlson-integral-RF 0 𝑦 𝑧 𝑦 𝑧 {\displaystyle{\displaystyle\ln 4\leq\sqrt{z}R_{F}\left(0,y,z\right)+\ln\sqrt{% y/z}}}
\ln@@{4} \leq \sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}}		 0 < y , y ≤ z formulae-sequence 0 𝑦 𝑦 𝑧 {\displaystyle{\displaystyle 0<y,y\leq z}} 
ln(4) <= sqrt(x + y*I)*0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)+ ln(sqrt(y/(x + y*I)))

Log[4] <= Sqrt[x + y*I]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+ Log[Sqrt[y/(x + y*I)]]
Error Failure -
Failed [9 / 9]
Result: LessEqual[1.3862943611198906, Complex[0.5672499697282593, -1.7874177081206242]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[1.3862943611198906, Complex[0.6277320470267476, -0.9602476282953896]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E1 z ⁢ R F ⁡ ( 0 , y , z ) + ln ⁡ y / z ≤ 1 2 ⁢ π 𝑧 Carlson-integral-RF 0 𝑦 𝑧 𝑦 𝑧 1 2 𝜋 {\displaystyle{\displaystyle\sqrt{z}R_{F}\left(0,y,z\right)+\ln\sqrt{y/z}\leq% \tfrac{1}{2}\pi}}
\sqrt{z}\CarlsonsymellintRF@{0}{y}{z}+\ln@@{\sqrt{y/z}} \leq \tfrac{1}{2}\pi		 0 < y , y ≤ z formulae-sequence 0 𝑦 𝑦 𝑧 {\displaystyle{\displaystyle 0<y,y\leq z}} 
sqrt(x + y*I)*0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)+ ln(sqrt(y/(x + y*I))) <= (1)/(2)*Pi

Sqrt[x + y*I]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+ Log[Sqrt[y/(x + y*I)]] <= Divide[1,2]*Pi
Error Failure -
Failed [9 / 9]
Result: LessEqual[Complex[0.5672499697282593, -1.7874177081206242], 1.5707963267948966]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[Complex[0.6277320470267476, -0.9602476282953896], 1.5707963267948966]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E2 1 2 ≤ z - 1 / 2 ⁢ R G ⁡ ( 0 , y , z ) 1 2 superscript 𝑧 1 2 Carlson-integral-RG 0 𝑦 𝑧 {\displaystyle{\displaystyle\tfrac{1}{2}\leq z^{-1/2}R_{G}\left(0,y,z\right)}}
\tfrac{1}{2} \leq z^{-1/2}\CarlsonsymellintRG@{0}{y}{z}		 0 ≤ y , y ≤ z formulae-sequence 0 𝑦 𝑦 𝑧 {\displaystyle{\displaystyle 0\leq y,y\leq z}} 
Error

Divide[1,2] <= (x + y*I)^(- 1/2)* Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2])
Missing Macro Error Failure -
Failed [9 / 9]
Result: LessEqual[0.5, Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[0.5, Plus[Complex[1.0897585107701309, 0.2919625251300463], Times[Complex[0.3515775842541431, 0.5688644810057831], Power[Plus[1.0, Times[Complex[-1.0, 0.5], Power[k, 2]]], Rational[1, 2]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E2 z - 1 / 2 ⁢ R G ⁡ ( 0 , y , z ) ≤ 1 4 ⁢ π superscript 𝑧 1 2 Carlson-integral-RG 0 𝑦 𝑧 1 4 𝜋 {\displaystyle{\displaystyle z^{-1/2}R_{G}\left(0,y,z\right)\leq\tfrac{1}{4}% \pi}}
z^{-1/2}\CarlsonsymellintRG@{0}{y}{z} \leq \tfrac{1}{4}\pi		 0 ≤ y , y ≤ z formulae-sequence 0 𝑦 𝑦 𝑧 {\displaystyle{\displaystyle 0\leq y,y\leq z}} 
Error

(x + y*I)^(- 1/2)* Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2]) <= Divide[1,4]*Pi
Missing Macro Error Failure -
Failed [9 / 9]
Result: LessEqual[Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]], 0.7853981633974483]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[Plus[Complex[1.0897585107701309, 0.2919625251300463], Times[Complex[0.3515775842541431, 0.5688644810057831], Power[Plus[1.0, Times[Complex[-1.0, 0.5], Power[k, 2]]], Rational[1, 2]]]], 0.7853981633974483]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E3 ( y 3 / 2 + z 3 / 2 2 ) 2 / 3 ≤ 4 π ⁢ R G ⁡ ( 0 , y 2 , z 2 ) superscript superscript 𝑦 3 2 superscript 𝑧 3 2 2 2 3 4 𝜋 Carlson-integral-RG 0 superscript 𝑦 2 superscript 𝑧 2 {\displaystyle{\displaystyle\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3}\leq% \frac{4}{\pi}R_{G}\left(0,y^{2},z^{2}\right)}}
\left(\frac{y^{3/2}+z^{3/2}}{2}\right)^{2/3} \leq \frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}}		 y > 0 , z > 0 formulae-sequence 𝑦 0 𝑧 0 {\displaystyle{\displaystyle y>0,z>0}} 
Error

(Divide[(y)^(3/2)+(x + y*I)^(3/2),2])^(2/3) <= Divide[4,Pi]*Sqrt[(x + y*I)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(x + y*I)^(2)]]]^2])
Missing Macro Error Failure -
Failed [9 / 9]
Result: LessEqual[Complex[1.4250443092558214, 0.7875512141675095], Complex[2.850438542245679, 1.5730146161508307]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[Complex[1.0588191704631045, 0.29794136993360365], Complex[2.118851869395612, 0.5983245902184247]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E3 4 π ⁢ R G ⁡ ( 0 , y 2 , z 2 ) ≤ ( y 2 + z 2 2 ) 1 / 2 4 𝜋 Carlson-integral-RG 0 superscript 𝑦 2 superscript 𝑧 2 superscript superscript 𝑦 2 superscript 𝑧 2 2 1 2 {\displaystyle{\displaystyle\frac{4}{\pi}R_{G}\left(0,y^{2},z^{2}\right)\leq% \left(\frac{y^{2}+z^{2}}{2}\right)^{1/2}}}
\frac{4}{\pi}\CarlsonsymellintRG@{0}{y^{2}}{z^{2}} \leq \left(\frac{y^{2}+z^{2}}{2}\right)^{1/2}		 y > 0 , z > 0 formulae-sequence 𝑦 0 𝑧 0 {\displaystyle{\displaystyle y>0,z>0}} 
Error

Divide[4,Pi]*Sqrt[(x + y*I)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(x + y*I)^(2)]],((x + y*I)^(2)-(y)^(2))/((x + y*I)^(2)-0)]+Cot[ArcCos[Sqrt[0/(x + y*I)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(x + y*I)^(2)]]]^2]) <= (Divide[(y)^(2)+(x + y*I)^(2),2])^(1/2)
Missing Macro Error Failure -
Failed [9 / 9]
Result: LessEqual[Complex[2.850438542245679, 1.5730146161508307], Complex[1.3491805799609005, 0.8338394553771318]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}


Result: LessEqual[Complex[2.118851869395612, 0.5983245902184247], Complex[1.112897508375995, 0.3369582528288897]]
Test Values: {Rule[x, 1.5], Rule[y, 0.5]}

... skip entries to safe data
19.24.E4 2 p ⁢ ( 2 ⁢ y ⁢ z + y ⁢ p + z ⁢ p ) - 1 / 2 ≤ 4 3 ⁢ π ⁢ R J ⁡ ( 0 , y , z , p ) 2 𝑝 superscript 2 𝑦 𝑧 𝑦 𝑝 𝑧 𝑝 1 2 4 3 𝜋 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2}\leq\frac{4}{3% \pi}R_{J}\left(0,y,z,p\right)}}
\frac{2}{\sqrt{p}}(2yz+yp+zp)^{-1/2} \leq \frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p}		 

Error

Divide[2,Sqrt[p]]*(2*y*(x + y*I)+ y*p +(x + y*I)*p)^(- 1/2) <= Divide[4,3*Pi]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0]
Missing Macro Error Failure -
Failed [180 / 180]
Result: LessEqual[Complex[0.13508456755677706, -1.1829936015765863], Complex[-0.3213270063391195, -0.3051912044731223]]
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]}


Result: LessEqual[Complex[0.7797231369520263, -0.6247258696161743], Complex[-0.6706782382611747, 0.54526856836685]]
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]}

... skip entries to safe data
19.24.E4 4 3 ⁢ π ⁢ R J ⁡ ( 0 , y , z , p ) ≤ ( y ⁢ z ⁢ p 2 ) - 3 / 8 4 3 𝜋 Carlson-integral-RJ 0 𝑦 𝑧 𝑝 superscript 𝑦 𝑧 superscript 𝑝 2 3 8 {\displaystyle{\displaystyle\frac{4}{3\pi}R_{J}\left(0,y,z,p\right)\leq(yzp^{2% })^{-3/8}}}
\frac{4}{3\pi}\CarlsonsymellintRJ@{0}{y}{z}{p} \leq (yzp^{2})^{-3/8}		 

Error

Divide[4,3*Pi]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] <= (y*(x + y*I)*(p)^(2))^(- 3/8)
Missing Macro Error Failure -
Failed [180 / 180]
Result: LessEqual[Complex[-0.3213270063391195, -0.3051912044731223], Complex[0.5136265917030035, 0.9609277658721954]]
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]}


Result: LessEqual[Complex[-0.6706782382611747, 0.54526856836685], Complex[0.8422602311268256, -0.6912251080442312]]
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]}

... skip entries to safe data
19.24.E5 1 a n ≤ 2 π ⁢ R F ⁡ ( 0 , a 0 2 , g 0 2 ) 1 subscript 𝑎 𝑛 2 𝜋 Carlson-integral-RF 0 superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 {\displaystyle{\displaystyle\frac{1}{a_{n}}\leq\frac{2}{\pi}R_{F}\left(0,a_{0}% ^{2},g_{0}^{2}\right)}}
\frac{1}{a_{n}} \leq \frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}}		 

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

Divide[1,Subscript[a, n]] <= Divide[2,Pi]*EllipticF[ArcCos[Sqrt[0/(Subscript[g, 0])^(2)]],((Subscript[g, 0])^(2)-(Subscript[a, 0])^(2))/((Subscript[g, 0])^(2)-0)]/Sqrt[(Subscript[g, 0])^(2)-0]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: LessEqual[Complex[1.7320508075688774, -0.9999999999999999], Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: LessEqual[Complex[1.7320508075688774, -0.9999999999999999], Times[2.0, Power[Times[Complex[-0.4999999999999998, -0.8660254037844387], g], Rational[-1, 2]], EllipticK[Times[Complex[-1.9999999999999991, 3.464101615137755], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[-0.12499999999999994, -0.21650635094610968], g]], Power[g, -1]]]]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.24.E5 2 π ⁢ R F ⁡ ( 0 , a 0 2 , g 0 2 ) ≤ 1 g n 2 𝜋 Carlson-integral-RF 0 superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 1 subscript 𝑔 𝑛 {\displaystyle{\displaystyle\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}% \right)\leq\frac{1}{g_{n}}}}
\frac{2}{\pi}\CarlsonsymellintRF@{0}{a_{0}^{2}}{g_{0}^{2}} \leq \frac{1}{g_{n}}		 

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

Divide[2,Pi]*EllipticF[ArcCos[Sqrt[0/(Subscript[g, 0])^(2)]],((Subscript[g, 0])^(2)-(Subscript[a, 0])^(2))/((Subscript[g, 0])^(2)-0)]/Sqrt[(Subscript[g, 0])^(2)-0] <= Divide[1,Subscript[g, n]]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: LessEqual[Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]], Complex[1.7320508075688774, -0.9999999999999999]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: LessEqual[Times[2.0, Power[Times[Complex[0.5000000000000001, 0.8660254037844386], g], Rational[-1, 2]], EllipticK[Times[Complex[2.0000000000000004, -3.4641016151377544], Plus[Times[Complex[-0.12500000000000003, -0.21650635094610965], a], Times[Complex[0.12500000000000003, 0.21650635094610965], g]], Power[g, -1]]]], Complex[-0.9999999999999996, -1.7320508075688774]]
Test Values: {Rule[n, 3], Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, n], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.24#Ex1 a n + 1 = ( a n + g n ) / 2 subscript 𝑎 𝑛 1 subscript 𝑎 𝑛 subscript 𝑔 𝑛 2 {\displaystyle{\displaystyle a_{n+1}=(a_{n}+g_{n})/2}}
a_{n+1} = (a_{n}+g_{n})/2		 

a[n + 1] = (a[n]+ g[n])/2
 
Subscript[a, n + 1] == (Subscript[a, n]+ Subscript[g, n])/2
 Skipped - no semantic math Skipped - no semantic math - -
19.24#Ex2 g n + 1 = a n ⁢ g n subscript 𝑔 𝑛 1 subscript 𝑎 𝑛 subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n+1}=\sqrt{a_{n}g_{n}}}}
g_{n+1} = \sqrt{a_{n}g_{n}}		 

g[n + 1] = sqrt(a[n]*g[n])
 
Subscript[g, n + 1] == Sqrt[Subscript[a, n]*Subscript[g, n]]
 Skipped - no semantic math Skipped - no semantic math - -
19.24.E7 L ⁢ ( a , b ) = 8 ⁢ R G ⁡ ( 0 , a 2 , b 2 ) 𝐿 𝑎 𝑏 8 Carlson-integral-RG 0 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle L(a,b)=8R_{G}\left(0,a^{2},b^{2}\right)}}
L(a,b) = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}		 

Error

L[a , b] == 8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2])
Missing Macro Error Failure - Error
19.24#Ex3 R F ⁡ ( x , y , 0 ) ⁢ R G ⁡ ( x , y , 0 ) > 1 8 ⁢ π 2 Carlson-integral-RF 𝑥 𝑦 0 Carlson-integral-RG 𝑥 𝑦 0 1 8 superscript 𝜋 2 {\displaystyle{\displaystyle R_{F}\left(x,y,0\right)R_{G}\left(x,y,0\right)>% \tfrac{1}{8}\pi^{2}}}
\CarlsonsymellintRF@{x}{y}{0}\CarlsonsymellintRG@{x}{y}{0} > \tfrac{1}{8}\pi^{2}		 

Error

EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]/Sqrt[0-x]*Sqrt[0-x]*(EllipticE[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+(Cot[ArcCos[Sqrt[x/0]]])^2*EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+Cot[ArcCos[Sqrt[x/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/0]]]^2]) > Divide[1,8]*(Pi)^(2)
Missing Macro Error Failure -
Failed [18 / 18]
Result: Greater[Indeterminate, 1.2337005501361697]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Greater[Indeterminate, 1.2337005501361697]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24#Ex4 R F ⁡ ( x , y , 0 ) + 2 ⁢ R G ⁡ ( x , y , 0 ) > π Carlson-integral-RF 𝑥 𝑦 0 2 Carlson-integral-RG 𝑥 𝑦 0 𝜋 {\displaystyle{\displaystyle R_{F}\left(x,y,0\right)+2R_{G}\left(x,y,0\right)>% \pi}}
\CarlsonsymellintRF@{x}{y}{0}+2\CarlsonsymellintRG@{x}{y}{0} > \pi		 

Error

EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]/Sqrt[0-x]+ 2*Sqrt[0-x]*(EllipticE[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+(Cot[ArcCos[Sqrt[x/0]]])^2*EllipticF[ArcCos[Sqrt[x/0]],(0-y)/(0-x)]+Cot[ArcCos[Sqrt[x/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/0]]]^2]) > Pi
Missing Macro Error Failure -
Failed [18 / 18]
Result: Greater[Indeterminate, 3.141592653589793]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Greater[Indeterminate, 3.141592653589793]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24.E9 1 2 ⁢ g 1 2 ≤ R G ⁡ ( a 0 2 , g 0 2 , 0 ) R F ⁡ ( a 0 2 , g 0 2 , 0 ) 1 2 superscript subscript 𝑔 1 2 Carlson-integral-RG superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 0 Carlson-integral-RF superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 0 {\displaystyle{\displaystyle\frac{1}{2}\,g_{1}^{2}\leq\frac{R_{G}\left(a_{0}^{% 2},g_{0}^{2},0\right)}{R_{F}\left(a_{0}^{2},g_{0}^{2},0\right)}}}
\frac{1}{2}\,g_{1}^{2} \leq \frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}}		 

Error

Divide[1,2]*(Subscript[g, 1])^(2) <= Divide[Sqrt[0-(Subscript[a, 0])^(2)]*(EllipticE[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+(Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]])^2*EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]^2]),EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]/Sqrt[0-(Subscript[a, 0])^(2)]]
Missing Macro Error Failure -
Failed [300 / 300]
Result: LessEqual[Complex[0.06250000000000001, 0.10825317547305482], Indeterminate]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: LessEqual[Complex[-0.06249999999999997, -0.10825317547305484], Indeterminate]
Test Values: {Rule[Subscript[a, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[g, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.24.E9 R G ⁡ ( a 0 2 , g 0 2 , 0 ) R F ⁡ ( a 0 2 , g 0 2 , 0 ) ≤ 1 2 ⁢ a 1 2 Carlson-integral-RG superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 0 Carlson-integral-RF superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 0 1 2 superscript subscript 𝑎 1 2 {\displaystyle{\displaystyle\frac{R_{G}\left(a_{0}^{2},g_{0}^{2},0\right)}{R_{% F}\left(a_{0}^{2},g_{0}^{2},0\right)}\leq\frac{1}{2}\,a_{1}^{2}}}
\frac{\CarlsonsymellintRG@{a_{0}^{2}}{g_{0}^{2}}{0}}{\CarlsonsymellintRF@{a_{0}^{2}}{g_{0}^{2}}{0}} \leq \frac{1}{2}\,a_{1}^{2}		 

Error

Divide[Sqrt[0-(Subscript[a, 0])^(2)]*(EllipticE[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+(Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]])^2*EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]+Cot[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]]]^2]),EllipticF[ArcCos[Sqrt[(Subscript[a, 0])^(2)/0]],(0-(Subscript[g, 0])^(2))/(0-(Subscript[a, 0])^(2))]/Sqrt[0-(Subscript[a, 0])^(2)]] <= Divide[1,2]*(Subscript[a, 1])^(2)
Missing Macro Error Failure -
Failed [300 / 300]
Result: LessEqual[Indeterminate, Complex[0.06250000000000001, 0.10825317547305482]]
Test Values: {Rule[Subscript[a, 0], 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[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: LessEqual[Indeterminate, Complex[0.06250000000000001, 0.10825317547305482]]
Test Values: {Rule[Subscript[a, 0], 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[g, 0], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.24.E10 3 x + y + z ≤ R F ⁡ ( x , y , z ) 3 𝑥 𝑦 𝑧 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\leq R_{F}% \left(x,y,z\right)}}
\frac{3}{\sqrt{x}+\sqrt{y}+\sqrt{z}} \leq \CarlsonsymellintRF@{x}{y}{z}		 

(3)/(sqrt(x)+sqrt(y)+sqrt(x + y*I)) <= 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)

Divide[3,Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]] <= EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
Aborted Failure Error
Failed [18 / 18]
Result: LessEqual[Complex[1.0934408788539995, -0.2839050517129825], Complex[-0.16214470973156064, 0.6784437678906974]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: LessEqual[Complex[0.7738030002696183, -0.11364498174072818], Complex[-0.28823404661462, -0.7809212115368181]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24.E10 R F ⁡ ( x , y , z ) ≤ 1 ( x ⁢ y ⁢ z ) 1 / 6 Carlson-integral-RF 𝑥 𝑦 𝑧 1 superscript 𝑥 𝑦 𝑧 1 6 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)\leq\frac{1}{(xyz)^{1/6}}}}
\CarlsonsymellintRF@{x}{y}{z} \leq \frac{1}{(xyz)^{1/6}}		 

0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) <= (1)/((x*y*(x + y*I))^(1/6))

EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] <= Divide[1,(x*y*(x + y*I))^(1/6)]
Aborted Failure Error
Failed [18 / 18]
Result: LessEqual[Complex[-0.16214470973156064, 0.6784437678906974], Complex[0.7120063770987297, -0.29492269789042613]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: LessEqual[Complex[-0.28823404661462, -0.7809212115368181], Complex[0.7640769591692358, -0.10059264002361257]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24.E11 ( 5 x + y + z + 2 ⁢ p ) 3 ≤ R J ⁡ ( x , y , z , p ) superscript 5 𝑥 𝑦 𝑧 2 𝑝 3 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p% }}\right)^{3}\leq R_{J}\left(x,y,z,p\right)}}
\left(\frac{5}{\sqrt{x}+\sqrt{y}+\sqrt{z}+2\sqrt{p}}\right)^{3} \leq \CarlsonsymellintRJ@{x}{y}{z}{p}		 

Error

(Divide[5,Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]+ 2*Sqrt[p]])^(3) <= 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]
Missing Macro Error Failure -
Failed [180 / 180]
Result: LessEqual[Complex[1.3310335634294785, -1.2911719373315522], Complex[-0.2876927312707393, -0.327259429717868]]
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]}


Result: LessEqual[Complex[0.7477899794343462, -0.4392695700678081], Complex[-0.36602768453446033, 0.5058947820270108]]
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]}

... skip entries to safe data
19.24.E11 R J ⁡ ( x , y , z , p ) ≤ ( x ⁢ y ⁢ z ⁢ p 2 ) - 3 / 10 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 superscript 𝑥 𝑦 𝑧 superscript 𝑝 2 3 10 {\displaystyle{\displaystyle R_{J}\left(x,y,z,p\right)\leq(xyzp^{2})^{-3/10}}}
\CarlsonsymellintRJ@{x}{y}{z}{p} \leq (xyzp^{2})^{-3/10}		 

Error

3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] <= (x*y*(x + y*I)*(p)^(2))^(- 3/10)
Missing Macro Error Failure -
Failed [180 / 180]
Result: LessEqual[Complex[-0.2876927312707393, -0.327259429717868], Complex[0.6159220908806466, 0.7211521128667333]]
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]}


Result: LessEqual[Complex[-0.36602768453446033, 0.5058947820270108], Complex[0.8086249764673956, -0.49552602288885395]]
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]}

... skip entries to safe data
19.24.E12 1 3 ⁢ ( x + y + z ) ≤ R G ⁡ ( x , y , z ) 1 3 𝑥 𝑦 𝑧 Carlson-integral-RG 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq R_{G}% \left(x,y,z\right)}}
\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z}) \leq \CarlsonsymellintRG@{x}{y}{z}		 

Error

Divide[1,3]*(Sqrt[x]+Sqrt[y]+Sqrt[x + y*I]) <= Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2])
Missing Macro Error Failure -
Failed [18 / 18]
Result: LessEqual[Complex[0.8567842015469013, 0.22245863288189585], Times[Complex[0.8660254037844386, -0.8660254037844385], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: LessEqual[Complex[1.2650324920107643, 0.1857896575819671], Times[Complex[0.8660254037844386, 0.8660254037844385], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24#Ex7 R F ⁡ ( x , y , z ) ⁢ R G ⁡ ( x , y , z ) > 1 Carlson-integral-RF 𝑥 𝑦 𝑧 Carlson-integral-RG 𝑥 𝑦 𝑧 1 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)R_{G}\left(x,y,z\right)>1}}
\CarlsonsymellintRF@{x}{y}{z}\CarlsonsymellintRG@{x}{y}{z} > 1		 

Error
 
EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]*Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) > 1
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Greater[Times[Complex[0.44712810031579164, 0.7279709757493625], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], 1.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Greater[Times[Complex[0.42667960094115687, -0.925915614148855], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]], 1.0]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24#Ex8 R F ⁡ ( x , y , z ) + R G ⁡ ( x , y , z ) > 2 Carlson-integral-RF 𝑥 𝑦 𝑧 Carlson-integral-RG 𝑥 𝑦 𝑧 2 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)+R_{G}\left(x,y,z\right)>2}}
\CarlsonsymellintRF@{x}{y}{z}+\CarlsonsymellintRG@{x}{y}{z} > 2		 

Error
 
EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]+ Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) > 2
 Missing Macro Error Failure -
Failed [18 / 18]
Result: Greater[Plus[Complex[-0.16214470973156064, 0.6784437678906974], Times[Complex[0.8660254037844386, -0.8660254037844385], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]], 2.0]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Greater[Plus[Complex[-0.28823404661462, -0.7809212115368181], Times[Complex[0.8660254037844386, 0.8660254037844385], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]], 2.0]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24.E15 R C ⁡ ( x , 1 2 ⁢ ( y + z ) ) ≤ R F ⁡ ( x , y , z ) Carlson-integral-RC 𝑥 1 2 𝑦 𝑧 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{C}\left(x,\tfrac{1}{2}(y+z)\right)\leq R_{F}% \left(x,y,z\right)}}
\CarlsonellintRC@{x}{\tfrac{1}{2}(y+z)} \leq \CarlsonsymellintRF@{x}{y}{z}		 x ≥ 0 𝑥 0 {\displaystyle{\displaystyle x\geq 0}} 
Error
 
1/Sqrt[Divide[1,2]*(y +(x + y*I))]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(Divide[1,2]*(y +(x + y*I)))] <= EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
 Missing Macro Error Failure -
Failed [18 / 18]
Result: LessEqual[Complex[0.9580693887321644, 0.49152363500125495], Complex[-0.16214470973156064, 0.6784437678906974]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: LessEqual[Complex[0.7805167095081702, -0.12346643314922054], Complex[-0.28823404661462, -0.7809212115368181]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.24.E15 R F ⁡ ( x , y , z ) ≤ R C ⁡ ( x , y ⁢ z ) Carlson-integral-RF 𝑥 𝑦 𝑧 Carlson-integral-RC 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)\leq R_{C}\left(x,\sqrt{yz% }\right)}}
\CarlsonsymellintRF@{x}{y}{z} \leq \CarlsonellintRC@{x}{\sqrt{yz}}		 x ≥ 0 𝑥 0 {\displaystyle{\displaystyle x\geq 0}} 
Error
 
EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] <= 1/Sqrt[Sqrt[y*(x + y*I)]]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(Sqrt[y*(x + y*I)])]
 Missing Macro Error Failure -
Failed [18 / 18]
Result: LessEqual[Complex[-0.16214470973156064, 0.6784437678906974], Complex[0.7308447207533646, -0.31118718328917466]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: LessEqual[Complex[-0.28823404661462, -0.7809212115368181], Complex[0.765857524311696, -0.1031964554328576]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
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