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| | <div style="-moz-column-count:2; column-count:2;"> |
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| | ; Notation : [[36.1|36.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Properties : [[36.2|36.2 Catastrophes and Canonical Integrals]]<br>[[36.3|36.3 Visualizations of Canonical Integrals]]<br>[[36.4|36.4 Bifurcation Sets]]<br>[[36.5|36.5 Stokes Sets]]<br>[[36.6|36.6 Scaling Relations]]<br>[[36.7|36.7 Zeros]]<br>[[36.8|36.8 Convergent Series Expansions]]<br>[[36.9|36.9 Integral Identities]]<br>[[36.10|36.10 Differential Equations]]<br>[[36.11|36.11 Leading-Order Asymptotics]]<br> |
| |-
| | ; Applications : [[36.12|36.12 Uniform Approximation of Integrals]]<br>[[36.13|36.13 Kelvin’s Ship-Wave Pattern]]<br>[[36.14|36.14 Other Physical Applications]]<br> |
| | [https://dlmf.nist.gov/36.2#Ex2 36.2#Ex2] || [[Item:Q9843|<math>\mathrm{F}_{+}(\mathbf{x}) = \int_{0}^{\infty}\cos@{ry\exp@{+ i\dfrac{\pi}{6}}}\exp@{2ir^{2}z\exp@{+ i\dfrac{\pi}{3}}}\AiryAi@{3^{2/3}r^{2}+3^{-1/3}\exp@{- i\dfrac{\pi}{3}}\left(\tfrac{1}{3}z^{2}-x\right)}\diff{r}</math>]] || <code>F[+]*(x) = int(cos(r*y*exp(+ I*(Pi)/(6)))*exp(2*I*(r)^(2)*(x + y*I)* exp(+ I*(Pi)/(3)))*AiryAi((3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* exp(- I*(Pi)/(3))*((1)/(3)*(x + y*I)^(2)- x)), r = 0..infinity)</code> || <code>Subscript[F, +]*(x) == Integrate[Cos[r*y*Exp[+ I*Divide[Pi,6]]]*Exp[2*I*(r)^(2)*(x + y*I)* Exp[+ I*Divide[Pi,3]]]*AiryAi[(3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* Exp[- I*Divide[Pi,3]]*(Divide[1,3]*(x + y*I)^(2)- x)], {r, 0, Infinity}, GenerateConditions->None]</code> || Error || Failure || - || Error
| | ; Computation : [[36.15|36.15 Methods of Computation]]<br> |
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| | </div> |
| | [https://dlmf.nist.gov/36.2#Ex2 36.2#Ex2] || [[Item:Q9843|<math>\mathrm{F}_{-}(\mathbf{x}) = \int_{0}^{\infty}\cos@{ry\exp@{- i\dfrac{\pi}{6}}}\exp@{2ir^{2}z\exp@{- i\dfrac{\pi}{3}}}\AiryAi@{3^{2/3}r^{2}+3^{-1/3}\exp@{+ i\dfrac{\pi}{3}}\left(\tfrac{1}{3}z^{2}-x\right)}\diff{r}</math>]] || <code>F[-]*(x) = int(cos(r*y*exp(- I*(Pi)/(6)))*exp(2*I*(r)^(2)*(x + y*I)* exp(- I*(Pi)/(3)))*AiryAi((3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* exp(+ I*(Pi)/(3))*((1)/(3)*(x + y*I)^(2)- x)), r = 0..infinity)</code> || <code>Subscript[F, -]*(x) == Integrate[Cos[r*y*Exp[- I*Divide[Pi,6]]]*Exp[2*I*(r)^(2)*(x + y*I)* Exp[- I*Divide[Pi,3]]]*AiryAi[(3)^(2/ 3)* (r)^(2)+ (3)^(- 1/ 3)* Exp[+ I*Divide[Pi,3]]*(Divide[1,3]*(x + y*I)^(2)- x)], {r, 0, Infinity}, GenerateConditions->None]</code> || Error || Failure || - || Error
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| | [https://dlmf.nist.gov/36.2.E14 36.2.E14] || [[Item:Q9850|<math>P(x_{2},x_{1}) = \int_{-\infty}^{\infty}\exp@{\iunit(t^{4}+x_{2}t^{2}+x_{1}t)}\diff{t}</math>]] || <code>P*(x[2], x[1]) = int(exp(I*((t)^(4)+ x[2]*(t)^(2)+ x[1]*t)), t = - infinity..infinity)</code> || <code>P*(Subscript[x, 2], Subscript[x, 1]) == Integrate[Exp[I*((t)^(4)+ Subscript[x, 2]*(t)^(2)+ Subscript[x, 1]*t)], {t, - Infinity, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Error
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| | [https://dlmf.nist.gov/36.2#Ex10 36.2#Ex10] || [[Item:Q9859|<math>\tfrac{1}{3}\sqrt{\pi}\EulerGamma@{\tfrac{1}{6}} = 3.28868</math>]] || <code>(1)/(3)*sqrt(Pi)*GAMMA((1)/(6)) = 3.28868</code> || <code>Divide[1,3]*Sqrt[Pi]*Gamma[Divide[1,6]] == 3.28868</code> || Failure || Failure || Successful [Tested: 0] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/36.2#Ex11 36.2#Ex11] || [[Item:Q9860|<math>\tfrac{1}{3}\EulerGamma^{2}@{\tfrac{1}{3}} = 2.39224</math>]] || <code>(1)/(3)*(GAMMA((1)/(3)))^(2) = 2.39224</code> || <code>Divide[1,3]*(Gamma[Divide[1,3]])^(2) == 2.39224</code> || Failure || Failure || Successful [Tested: 0] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/36.4.E5 36.4.E5] || [[Item:Q9877|<math>x = 0</math>]] || <code>x = 0</code> || <code>x == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4.E6 36.4.E6] || [[Item:Q9878|<math>27x^{2} = -8y^{3}</math>]] || <code>27*(x)^(2) = - 8*(y)^(3)</code> || <code>27*(x)^(2) == - 8*(y)^(3)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex3 36.4#Ex3] || [[Item:Q9879|<math>x = 3t^{2}(z+5t^{2})</math>]] || <code>x = 3*(t)^(2)*((x + y*I)+ 5*(t)^(2))</code> || <code>x == 3*(t)^(2)*((x + y*I)+ 5*(t)^(2))</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex4 36.4#Ex4] || [[Item:Q9880|<math>y = -t(3z+10t^{2})</math>]] || <code>y = - t*(3*(x + y*I)+ 10*(t)^(2))</code> || <code>y == - t*(3*(x + y*I)+ 10*(t)^(2))</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex5 36.4#Ex5] || [[Item:Q9881|<math>y = 0</math>]] || <code>y = 0</code> || <code>y == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex6 36.4#Ex6] || [[Item:Q9882|<math>z \leq 0</math>]] || <code>z <= 0</code> || <code>z <= 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex7 36.4#Ex7] || [[Item:Q9883|<math>x = \tfrac{9}{20}z^{2}</math>]] || <code>x = (9)/(20)*(x + y*I)^(2)</code> || <code>x == Divide[9,20]*(x + y*I)^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex8 36.4#Ex8] || [[Item:Q9884|<math>z \leq 0</math>]] || <code>z <= 0</code> || <code>z <= 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex9 36.4#Ex9] || [[Item:Q9885|<math>x = -\tfrac{3}{20}z^{2}</math>]] || <code>x = -(3)/(20)*(x + y*I)^(2)</code> || <code>x == -Divide[3,20]*(x + y*I)^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex10 36.4#Ex10] || [[Item:Q9886|<math>10y^{2} = -4z^{3}</math>]] || <code>10*(y)^(2) = - 4*(x + y*I)^(3)</code> || <code>10*(y)^(2) == - 4*(x + y*I)^(3)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.4#Ex11 36.4#Ex11] || [[Item:Q9887|<math>x = \tfrac{1}{3}z^{2}(-\cos@{2\phi}-2\cos@@{\phi})</math>]] || <code>x = (1)/(3)*(x + y*I)^(2)*(- cos(2*phi)- 2*cos(phi))</code> || <code>x == Divide[1,3]*(x + y*I)^(2)*(- Cos[2*\[Phi]]- 2*Cos[\[Phi]])</code> || Failure || Failure || <div class=" toccolours mw-collapsible mw- collapsed">Failed [180 / 180]<div class="mw- collapsible- content"><code>180/180]: [[-1.430780988-1.820001528*I <- {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>4.430780988+1.820001528*I <- {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2 }</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>{Complex[-1.4307809877134479, -1.8200015289703178] <- {Rule[x, 1.5], Rule[y, - 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[6.36271355956067, -6.045425917929804] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div> | |
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| | [https://dlmf.nist.gov/36.4#Ex12 36.4#Ex12] || [[Item:Q9888|<math>y = \tfrac{1}{3}z^{2}(\sin@{2\phi}-2\sin@@{\phi})</math>]] || <code>y = (1)/(3)*(x + y*I)^(2)*(sin(2*phi)- 2*sin(phi))</code> || <code>y == Divide[1,3]*(x + y*I)^(2)*(Sin[2*\[Phi]]- 2*Sin[\[Phi]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><code>54/54]: [[-1.500000000-2.780804948*I <- {phi = 3/2, x = 3/2, y = -3/2}</code><br><code>1.500000000+2.780804948*I <- {phi = 3/2, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><code>{Complex[-1.5, -2.7808049477223626] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, 1.5]}</code><br><code>Complex[-1.5, -0.17607013860076426] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[ϕ, 0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.4.E11 36.4.E11] || [[Item:Q9889|<math>x+iy = -z^{2}\exp@{\tfrac{2}{3}i\pi m}</math>]] || <code>x + I*y = -(x + y*I)^(2)* exp((2)/(3)*I*Pi*m)</code> || <code>x + I*y == -(x + y*I)^(2)* Exp[Divide[2,3]*I*Pi*m]</code> || Error || Failure || - || Skip - No test values generated
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| | [https://dlmf.nist.gov/36.4#Ex13 36.4#Ex13] || [[Item:Q9890|<math>x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}+ 2\exp@{-\tau})</math>]] || <code>x = -(1)/(12)*(x + y*I)^(2)*(exp(2*tau)+ 2*exp(- tau))</code> || <code>x == -Divide[1,12]*(x + y*I)^(2)*(Exp[2*\[Tau]]+ 2*Exp[- \[Tau]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>180/180]: [[3.132329502-1.422064683*I <- {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>-.132329502+1.422064683*I <- {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>{Complex[3.1323295016161525, -1.4220646828050367] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.694217640570763, -0.7789550680773418] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.4#Ex13 36.4#Ex13] || [[Item:Q9890|<math>x = -\tfrac{1}{12}z^{2}(\exp@{2\tau}- 2\exp@{-\tau})</math>]] || <code>x = -(1)/(12)*(x + y*I)^(2)*(exp(2*tau)- 2*exp(- tau))</code> || <code>x == -Divide[1,12]*(x + y*I)^(2)*(Exp[2*\[Tau]]- 2*Exp[- \[Tau]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>180/180]: [[3.434813476-.8683714834*I <- {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>-.434813476+.8683714834*I <- {tau = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>{Complex[3.4348134754242143, -0.8683714827345889] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[2.578112467364518, 0.8232541553428006] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.4#Ex14 36.4#Ex14] || [[Item:Q9891|<math>y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}+ 2\exp@{\tau})</math>]] || <code>y = -(1)/(12)*(x + y*I)^(2)*(exp(- 2*tau)+ 2*exp(tau))</code> || <code>y == -Divide[1,12]*(x + y*I)^(2)*(Exp[- 2*\[Tau]]+ 2*Exp[\[Tau]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [108 / 108]<div class="mw-collapsible-content"><code>108/108]: [[-1.500000000-7.699423965*I <- {tau = -3/2, x = 3/2, y = -3/2}</code><br><code>1.500000000+7.699423965*I <- {tau = -3/2, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [108 / 108]<div class="mw-collapsible-content"><code>{Complex[-1.5, -7.699423966306697] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, -1.5]}</code><br><code>Complex[-1.5, -3.3799369533914976] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.4#Ex14 36.4#Ex14] || [[Item:Q9891|<math>y = -\tfrac{1}{12}z^{2}(\exp@{-2\tau}- 2\exp@{\tau})</math>]] || <code>y = -(1)/(12)*(x + y*I)^(2)*(exp(- 2*tau)- 2*exp(tau))</code> || <code>y == -Divide[1,12]*(x + y*I)^(2)*(Exp[- 2*\[Tau]]- 2*Exp[\[Tau]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [108 / 108]<div class="mw-collapsible-content"><code>108/108]: [[-1.500000000-7.364728725*I <- {tau = -3/2, x = 3/2, y = -3/2}</code><br><code>1.500000000+7.364728725*I <- {tau = -3/2, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [108 / 108]<div class="mw-collapsible-content"><code>{Complex[-1.5, -7.364728726084054] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, -1.5]}</code><br><code>Complex[-1.5, 3.3425966521155988] <- {Rule[x, 1.5], Rule[y, -1.5], Rule[τ, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.4.E13 36.4.E13] || [[Item:Q9892|<math>x = y</math>]] || <code>x = y</code> || <code>x == y</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5.E2 36.5.E2] || [[Item:Q9895|<math>y^{3} = \tfrac{27}{4}\left(\sqrt{27}-5\right)x^{2}</math>]] || <code>(y)^(3) = (27)/(4)*(sqrt(27)- 5)* (x)^(2)</code> || <code>(y)^(3) == Divide[27,4]*(Sqrt[27]- 5)* (x)^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5#Ex3 36.5#Ex3] || [[Item:Q9896|<math>x = B_{+}|y|^{4/3}</math>]] || <code>x = B[+]*(abs(y))^(4/ 3)</code> || <code>x == Subscript[B, +]*(Abs[y])^(4/ 3)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5#Ex4 36.5#Ex4] || [[Item:Q9897|<math>B_{+} = 10^{-1/3}\left(2x_{+}^{4/3}-\tfrac{1}{2}x_{+}^{-2/3}\right)</math>]] || <code>B[+] (2*x(x[+])^(4/ 3)-(1)/(2)*x(x[+])^(- 2/ 3))</code> || <code>Subscript[B, +] (2*x(Subscript[x, +])^(4/ 3)-Divide[1,2]*x(Subscript[x, +])^(- 2/ 3))</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5.E4 36.5.E4] || [[Item:Q9898|<math>80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1 = 0</math>]] || <code>80*(x)^(5)- 40*(x)^(4)- 55*(x)^(3)+ 5*(x)^(2)+ 20*x - 1 = 0</code> || <code>80*(x)^(5)- 40*(x)^(4)- 55*(x)^(3)+ 5*(x)^(2)+ 20*x - 1 == 0</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5#Ex5 36.5#Ex5] || [[Item:Q9899|<math>B_{-} = -1.69916</math>]] || <code>B[-] = - 1.69916</code> || <code>Subscript[B, -] == - 1.69916</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5#Ex6 36.5#Ex6] || [[Item:Q9900|<math>B_{+} = 0.33912</math>]] || <code>B[+] = 0.33912</code> || <code>Subscript[B, +] == 0.33912</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/36.5.E7 36.5.E7] || [[Item:Q9903|<math>X = \dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\sign@{z}</math>]] || <code>X = (9)/(20)+ 20*(u)^(4)-((Y)^(2))/(20*(u)^(2))+ 6*(u)^(2)* signum(z)</code> || <code>X == Divide[9,20]+ 20*(u)^(4)-Divide[(Y)^(2),20*(u)^(2)]+ 6*(u)^(2)* Sign[z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[4.626363000-18.38362857*I <- {X = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>11.97483223-22.62626926*I <- {X = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[4.626363002559913, -18.383628553565494] <- {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[X, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}</code><br><code>Complex[11.974832230909447, -22.62626924068478] <- {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[X, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.5.E8 36.5.E8] || [[Item:Q9904|<math>16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\sign@{z}-\frac{3}{10}|Y|\sign@{z}+4t^{5}+2t^{3}\sign@{z}+|Y|t^{2} = 0</math>]] || <code>16*(u)^(5)-((Y)^(2))/(10*u)+ 4*(u)^(3)* signum(z)-(3)/(10)*abs(Y)* signum(z)+ 4*(t)^(5)+ 2*(t)^(3)* signum(z)+abs(Y)* (t)^(2) = 0</code> || <code>16*(u)^(5)-Divide[(Y)^(2),10*u]+ 4*(u)^(3)* Sign[z]-Divide[3,10]*Abs[Y]* Sign[z]+ 4*(t)^(5)+ 2*(t)^(3)* Sign[z]+Abs[Y]* (t)^(2) == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-45.20699439+15.71617138*I <- {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>-50.86384865+5.964253129*I <- {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-45.20699439419109, 15.716171367970516] <- {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}</code><br><code>Complex[-50.863848643683475, 5.964253107561413] <- {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, 1.5]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E9 36.5.E9] || [[Item:Q9905|<math>t = -u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}\sign@{z}\right)^{1/2}</math>]] || <code>t = - u +((abs(Y))/(10*u)- (u)^(2)-(3)/(10)*signum(z))^(1/ 2)</code> || <code>t == - u +(Divide[Abs[Y],10*u]- (u)^(2)-Divide[3,10]*Sign[z])^(1/ 2)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-1.010332328+1.400961906*I <- {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = -3/2}</code><br><code>-1.190759425+1.489998263*I <- {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, y = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.0103323276990124, 1.4009619057655258] <- {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5]}</code><br><code>Complex[-1.1907594253432556, 1.489998262392432] <- {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, 1.5]}</code><br></div></div> | |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E10 36.5.E10] || [[Item:Q9906|<math>160u^{6}+40u^{4} = Y^{2}</math>]] || <code>160*(u)^(6)+ 40*(u)^(4) = (Y)^(2)</code> || <code>160*(u)^(6)+ 40*(u)^(4) == (Y)^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E11 36.5.E11] || [[Item:Q9907|<math>\frac{x}{z^{2}} = -1-12u^{2}+8u-\left|\frac{y}{z^{2}}\right|\dfrac{\frac{1}{3}-u}{\left(u\left(\frac{2}{3}-u\right)\right)^{1/2}}</math>]] || <code>(x)/((x + y*I)^(2)) = - 1 - 12*(u)^(2)+ 8*u -abs((y)/((x + y*I)^(2)))*((1)/(3)- u)/((u*((2)/(3)- u))^(1/ 2))</code> || <code>Divide[x,(x + y*I)^(2)] == - 1 - 12*(u)^(2)+ 8*u -Abs[Divide[y,(x + y*I)^(2)]]*Divide[Divide[1,3]- u,(u*(Divide[2,3]- u))^(1/ 2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || - | |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E12 36.5.E12] || [[Item:Q9908|<math>8u^{3}-4u^{2}-\left|\frac{y}{3z^{2}}\right|\left(\frac{u}{\tfrac{2}{3}-u}\right)^{1/2} = \frac{y^{2}}{6wz^{4}}-2w^{3}-2w^{2}</math>]] || <code>8*(u)^(3)- 4*(u)^(2)-abs((y)/(3*(x + y*I)^(2)))*((u)/((2)/(3)- u))^(1/ 2) = ((y)^(2))/(6*w*(x + y*I)^(4))- 2*(w)^(3)- 2*(w)^(2)</code> || <code>8*(u)^(3)- 4*(u)^(2)-Abs[Divide[y,3*(x + y*I)^(2)]]*(Divide[u,Divide[2,3]- u])^(1/ 2) == Divide[(y)^(2),6*w*(x + y*I)^(4)]- 2*(w)^(3)- 2*(w)^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E14 36.5.E14] || [[Item:Q9910|<math>0 < u</math>]] || <code>0 < u</code> || <code>0 < u</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5#Ex11 36.5#Ex11] || [[Item:Q9913|<math>Y(u,X) = 8u-24u^{2}+X\dfrac{u-\tfrac{1}{6}}{\left(u\left(u-\tfrac{1}{3}\right)\right)^{1/2}}</math>]] || <code>Y*(u , X) = 8*u - 24*(u)^(2)+ X*(u -(1)/(6))/((u*(u -(1)/(3)))^(1/ 2))</code> || <code>Y*(u , X) == 8*u - 24*(u)^(2)+ X*Divide[u -Divide[1,6],(u*(u -Divide[1,3]))^(1/ 2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || - | |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E17 36.5.E17] || [[Item:Q9915|<math>Y_{\mathrm{S}}(X) = Y(u,|X|)</math>]] || <code>Y[S]*(X) = Y*(u ,abs(X))</code> || <code>Subscript[Y, S]*(X) == Y*(u ,Abs[X])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E19 36.5.E19] || [[Item:Q9917|<math>Y_{\mathrm{S}}(X) = Y(-u,-|X|)</math>]] || <code>Y[S]*(X) = Y*(- u , -abs(X))</code> || <code>Subscript[Y, S]*(X) == Y*(- u , -Abs[X])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.5.E21 36.5.E21] || [[Item:Q9919|<math>w = (\tfrac{1}{3}+u)\left(1-\left(1-\dfrac{|X|}{12u^{1/2}(\tfrac{1}{3}+u)^{3/2}}\right)^{1/2}\right)</math>]] || <code>w = ((1)/(3)+ u)*(1 -(1 -(abs(X))/(12*(u)^(1/ 2)*((1)/(3)+ u)^(3/ 2)))^(1/ 2))</code> || <code>w == (Divide[1,3]+ u)*(1 -(1 -Divide[Abs[X],12*(u)^(1/ 2)*(Divide[1,3]+ u)^(3/ 2)])^(1/ 2))</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.6#Ex9 36.6#Ex9] || [[Item:Q9928|<math>\gamma_{y}^{(\mathrm{U})} = \tfrac{2}{3}</math>]] || <code>(gamma[y])^(U) = (2)/(3)</code> || <code>(Subscript[\[Gamma], y])^(U) == Divide[2,3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.6#Ex10 36.6#Ex10] || [[Item:Q9929|<math>\gamma_{z}^{(\mathrm{U})} = \tfrac{1}{3}</math>]] || <code>(gamma[z])^(U) = (1)/(3)</code> || <code>(Subscript[\[Gamma], z])^(U) == Divide[1,3]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex1 36.7#Ex1] || [[Item:Q9932|<math>y_{m} = -\sqrt{2\pi(2m+1)}</math>]] || <code>y[m] = -sqrt(2*Pi*(2*m + 1))</code> || <code>Subscript[y, m] == -Sqrt[2*Pi*(2*m + 1)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex2 36.7#Ex2] || [[Item:Q9933|<math>x_{m,n}^{+} = \sqrt{\dfrac{2}{-y_{m}}}\left(2n+\tfrac{1}{2}+(-1)^{m}\tfrac{1}{2}+\tfrac{1}{4}\right)\pi</math>]] || <code>(x[m , n])^(+) = sqrt((2)/(- y[m]))*(2*n +(1)/(2)+(- 1)^(m)*(1)/(2)+(1)/(4))* Pi</code> || <code>(Subscript[x, m , n])^(+) == Sqrt[Divide[2,- Subscript[y, m]]]*(2*n +Divide[1,2]+(- 1)^(m)*Divide[1,2]+Divide[1,4])* Pi</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex3 36.7#Ex3] || [[Item:Q9934|<math>x_{n} = +\left(\dfrac{8}{27}\right)^{1/2}|y_{n}|^{3/2}(1+\xi_{n})</math>]] || <code>x[n] = +((8)/(27))^(1/ 2)*(abs(y[n]))^(3/ 2)*(1 + xi[n])</code> || <code>Subscript[x, n] == +(Divide[8,27])^(1/ 2)*(Abs[Subscript[y, n]])^(3/ 2)*(1 + Subscript[\[Xi], n])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex4 36.7#Ex4] || [[Item:Q9935|<math>y_{n} = -\left(\frac{3\pi(8n+5)}{9+8\xi_{n}}\right)^{1/2}</math>]] || <code>y[n] = -((3*Pi*(8*n + 5))/(9 + 8*xi[n]))^(1/ 2)</code> || <code>Subscript[y, n] == -(Divide[3*Pi*(8*n + 5),9 + 8*Subscript[\[Xi], n]])^(1/ 2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7.E3 36.7.E3] || [[Item:Q9936|<math>\frac{3\pi(8n+5)}{9+8\xi_{n}}\xi_{n}^{3/2} = \dfrac{27}{16}\left(\dfrac{3}{2}\right)^{1/2}\left(\ln@{\frac{1}{\xi_{n}}}+3\ln@{\dfrac{3}{2}}\right)</math>]] || <code>(3*Pi*(8*n + 5))/(9 + 8*xi[n])*(xi[n])^(3/ 2) = (27)/(16)*((3)/(2))^(1/ 2)*(ln((1)/(xi[n]))+ 3*ln((3)/(2)))</code> || <code>Divide[3*Pi*(8*n + 5),9 + 8*Subscript[\[Xi], n]]*(Subscript[\[Xi], n])^(3/ 2) == Divide[27,16]*(Divide[3,2])^(1/ 2)*(Log[Divide[1,Subscript[\[Xi], n]]]+ 3*Log[Divide[3,2]])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[3.887397376+4.913760852*I <- {xi = 1/2*3^(1/2)+1/2*I, xi[n] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>7.826714845+7.271674350*I <- {xi = 1/2*3^(1/2)+1/2*I, xi[n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[3.8873973728456934, 4.913760851775014] <- {Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[7.826714842534605, 7.271674348744825] <- {Rule[n, 2], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div> | |
| |-
| |
| | [https://dlmf.nist.gov/36.7.E4 36.7.E4] || [[Item:Q9937|<math>z_{n} = + 3(\tfrac{1}{4}\pi(2n-\tfrac{1}{2}))^{1/3}\\</math>]] || <code>z[n] = + 3*((1)/(4)*Pi*(2*n -(1)/(2)))^(1/ 3)</code> || <code>Subscript[z, n] == + 3*(Divide[1,4]*Pi*(2*n -Divide[1,2]))^(1/ 3)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex5 36.7#Ex5] || [[Item:Q9938|<math>\Delta z = \frac{9\pi}{2z_{n}^{2}}</math>]] || <code>Delta*z (9*Pi)/(2*z(z[n])^(2))</code> || <code>\[CapitalDelta]*z Divide[9*Pi,2*z(Subscript[z, n])^(2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || - | |
| |-
| |
| | [https://dlmf.nist.gov/36.7#Ex6 36.7#Ex6] || [[Item:Q9939|<math>\Delta x = \frac{6\pi}{z_{n}}</math>]] || <code>Delta*x = (6*Pi)/(x + y*I[n])</code> || <code>\[CapitalDelta]*x == Divide[6*Pi,Subscript[x + y*I, n]]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.7.E6 36.7.E6] || [[Item:Q9940|<math>\exp@{-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)}\*{\left(2\exp@{\frac{-6\pi ix}{\Delta x}}\cos@{\frac{2\sqrt{3}\pi y}{\Delta x}}+1\right)} = \sqrt{3}</math>]] || <code>exp(- 2*Pi*I*(((x + y*I)-x + y*I[n])/(Delta*(x + y*I))+(2*x)/(Delta*x)))*(2*exp((- 6*Pi*I*x)/(Delta*x))*cos((2*sqrt(3)*Pi*y)/(Delta*x))+ 1) = sqrt(3)</code> || <code>Exp[- 2*Pi*I*(Divide[(x + y*I)-Subscript[x + y*I, n],\[CapitalDelta]*(x + y*I)]+Divide[2*x,\[CapitalDelta]*x])]*(2*Exp[Divide[- 6*Pi*I*x,\[CapitalDelta]*x]]*Cos[Divide[2*Sqrt[3]*Pi*y,\[CapitalDelta]*x]]+ 1) == Sqrt[3]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Plus[-1.7320508075688772, Times[Complex[1.0151974851172445, -0.010763380729874927], Power[2.718281828459045, Times[Complex[0.0, -6.283185307179586], Plus[Complex[1.7320508075688774, -0.9999999999999999], Times[Complex[0.4553418012614795, 0.12200846792814624], Plus[Complex[1.5, -1.5], Times[-1.0, Subscript[Complex[1.5, -1.5], 1]]]]]]]]] <- {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[Δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[-1.7320508075688772, Times[Complex[1.0151974851172445, -0.010763380729874927], Power[2.718281828459045, Times[Complex[0.0, -6.283185307179586], Plus[Complex[1.7320508075688774, -0.9999999999999999], Times[Complex[0.4553418012614795, 0.12200846792814624], Plus[Complex[1.5, -1.5], Times[-1.0, Subscript[Complex[1.5, -1.5], 2]]]]]]]]] <- {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[Δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div> | |
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| |
| | [https://dlmf.nist.gov/36.9.E2 36.9.E2] || [[Item:Q9955|<math>(\AiryAi@{x})^{2} = \frac{2^{2/3}}{\pi}\int_{0}^{\infty}\AiryAi@{2^{2/3}(u^{2}+x)}\diff{u}</math>]] || <code>(AiryAi(x))^(2) = ((2)^(2/ 3))/(Pi)*int(AiryAi((2)^(2/ 3)*((u)^(2)+ x)), u = 0..infinity)</code> || <code>(AiryAi[x])^(2) == Divide[(2)^(2/ 3),Pi]*Integrate[AiryAi[(2)^(2/ 3)*((u)^(2)+ x)], {u, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| |
| |-
| |
| | [https://dlmf.nist.gov/36.11.E1 36.11.E1] || [[Item:Q9984|<math>t_{1}(\mathbf{x}) < t_{2}(\mathbf{x})</math>]] || <code>t[1]*(x) < t[2]*(x)</code> || <code>Subscript[t, 1]*(x) < Subscript[t, 2]*(x)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.12.E9 36.12.E9] || [[Item:Q10001|<math>P_{mn}(\mathbf{y}) = (t_{n}(\mathbf{x}(\mathbf{y})))^{K+1}+\sum_{l=m+2}^{K}\frac{l}{K+2}x_{l}(\mathbf{y})(t_{n}(\mathbf{x}(\mathbf{y})))^{l-1}</math>]] || <code>P[m*n]*(y) = (t[n]*(x*(y)))^(K + 1)+ sum((l)/(K + 2)*x[l]*(y)*(t[n]*(x*(y)))^(l - 1), l = m + 2..K)</code> || <code>Subscript[P, m*n]*(y) == (Subscript[t, n]*(x*(y)))^(K + 1)+ Sum[Divide[l,K + 2]*Subscript[x, l]*(y)*(Subscript[t, n]*(x*(y)))^(l - 1), {l, m + 2, K}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
| |
| |-
| |
| | [https://dlmf.nist.gov/36.13.E1 36.13.E1] || [[Item:Q10008|<math>z(\phi,\rho) = \int_{-\pi/2}^{\pi/2}\cos@{\rho\frac{\cos@{\theta+\phi}}{\cos^{2}@@{\theta}}}\diff{\theta}</math>]] || <code>z*(phi , rho) = int(cos(rho*(cos(theta + phi))/((cos(theta))^(2))), theta = - Pi/ 2..Pi/ 2)</code> || <code>z*(\[Phi], \[Rho]) == Integrate[Cos[\[Rho]*Divide[Cos[\[Theta]+ \[Phi]],(Cos[\[Theta]])^(2)]], {\[Theta], - Pi/ 2, Pi/ 2}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || Error
| |
| |-
| |
| | [https://dlmf.nist.gov/36.13.E2 36.13.E2] || [[Item:Q10009|<math>\rho = \ifrac{gr}{V^{2}}</math>]] || <code>rho = (g*r)/((V)^(2))</code> || <code>\[Rho] == Divide[g*r,(V)^(2)]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |-
| |
| | [https://dlmf.nist.gov/36.13#Ex1 36.13#Ex1] || [[Item:Q10011|<math>\theta_{+}(\phi) = \tfrac{1}{2}(\asin@{3\sin@@{\phi}}-\phi)</math>]] || <code>theta[+]*(phi) = (1)/(2)*(arcsin(3*sin(phi))- phi)</code> || <code>Subscript[\[Theta], +]*(\[Phi]) == Divide[1,2]*(ArcSin[3*Sin[\[Phi]]]- \[Phi])</code> || Error || Failure || - || Error
| |
| |-
| |
| | [https://dlmf.nist.gov/36.13#Ex2 36.13#Ex2] || [[Item:Q10012|<math>\theta_{-}(\phi) = \tfrac{1}{2}(\pi-\phi-\asin@{3\sin@@{\phi}})</math>]] || <code>theta[-]*(phi) = (1)/(2)*(Pi - phi - arcsin(3*sin(phi)))</code> || <code>Subscript[\[Theta], -]*(\[Phi]) == Divide[1,2]*(Pi - \[Phi]- ArcSin[3*Sin[\[Phi]]])</code> || Error || Failure || - || Error
| |
| |-
| |
| | [https://dlmf.nist.gov/36.13.E5 36.13.E5] || [[Item:Q10013|<math>|\phi| = \phi_{c}</math>]] || <code>abs(phi) = phi[c]</code> || <code>Abs[\[Phi]] == Subscript[\[Phi], c]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [282 / 300]<div class="mw-collapsible-content"><code>282/300]: [[.1339745960-.5000000000*I <- {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = 1/2*3^(1/2)+1/2*I}</code><br><code>1.500000000-.8660254040*I <- {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [282 / 300]<div class="mw-collapsible-content"><code>{Complex[0.1339745962155613, -0.49999999999999994] <- {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.4999999999999998, -0.8660254037844387] <- {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div> | |
| |-
| |
| | [https://dlmf.nist.gov/36.13.E5 36.13.E5] || [[Item:Q10013|<math>\phi_{c} = \asin@{\tfrac{1}{3}}</math>]] || <code>phi[c] = arcsin((1)/(3))</code> || <code>Subscript[\[Phi], c] == ArcSin[Divide[1,3]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.5261884946+.5000000000*I <- {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = 1/2*3^(1/2)+1/2*I}</code><br><code>-.8398369094+.8660254040*I <- {c = -3/2, phi = 1/2*3^(1/2)+1/2*I, phi[c] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.5261884943303168, 0.49999999999999994] <- {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.8398369094541217, 0.8660254037844387] <- {Rule[c, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, c], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/36.13.E6 36.13.E6] || [[Item:Q10014|<math>\omega(\mathbf{k}) = \sqrt{gk}+\mathbf{V}\cdot\mathbf{k}</math>]] || <code>omega*(k) = sqrt(g*k)+ V * k</code> || <code>\[Omega]*(k) == Sqrt[g*k]+ V * k</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |}
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