Heun Functions - 32.2 Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.2.E1 d 2 w d z 2 = 6 ā¢ w 2 + z derivative š‘¤ š‘§ 2 6 superscript š‘¤ 2 š‘§ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=6w^{2}% +z}}
\deriv[2]{w}{z} = 6w^{2}+z		 

diff(w, [z$(2)]) = 6*(w)^(2)+ z

D[w, {z, 2}] == 6*(w)^(2)+ z
Failure Failure
Failed [70 / 70]
Result: -3.866025406-5.696152424*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -2.500000002-6.062177828*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[-3.8660254037844397, -5.696152422706632]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.500000000000001, -6.06217782649107]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E2 d 2 w d z 2 = 2 ā¢ w 3 + z ā¢ w + Ī± derivative š‘¤ š‘§ 2 2 superscript š‘¤ 3 š‘§ š‘¤ š›¼ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=2w^{3}% +zw+\alpha}}
\deriv[2]{w}{z} = 2w^{3}+zw+\alpha		 

diff(w, [z$(2)]) = 2*(w)^(3)+ z*w + alpha

D[w, {z, 2}] == 2*(w)^(3)+ z*w + \[Alpha]
Failure Failure
Failed [209 / 210]
Result: -2.000000001-2.866025406*I
Test Values: {alpha = 3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.6339745966-2.500000002*I
Test Values: {alpha = 3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [209 / 210]
Result: Complex[-2.0, -2.8660254037844384]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5]}


Result: Complex[-1.0, -2.8660254037844384]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 0.5]}

... skip entries to safe data
32.2.E3 d 2 w d z 2 = 1 w ā¢ ( d w d z ) 2 - 1 z ā¢ d w d z + Ī± ā¢ w 2 + Ī² z + Ī³ ā¢ w 3 + Ī“ w derivative š‘¤ š‘§ 2 1 š‘¤ superscript derivative š‘¤ š‘§ 2 1 š‘§ derivative š‘¤ š‘§ š›¼ superscript š‘¤ 2 š›½ š‘§ š›¾ superscript š‘¤ 3 š›æ š‘¤ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{w}\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-\frac{1}{z}\frac{\mathrm% {d}w}{\mathrm{d}z}+\frac{\alpha w^{2}+\beta}{z}+\gamma w^{3}+\frac{\delta}{w}}}
\deriv[2]{w}{z} = \frac{1}{w}\left(\deriv{w}{z}\right)^{2}-\frac{1}{z}\deriv{w}{z}+\frac{\alpha w^{2}+\beta}{z}+\gamma w^{3}+\frac{\delta}{w}		 

diff(w, [z$(2)]) = (1)/(w)*(diff(w, z))^(2)-(1)/(z)*diff(w, z)+(alpha*(w)^(2)+ beta)/(z)+ gamma*(w)^(3)+(delta)/(w)

D[w, {z, 2}] == Divide[1,w]*(D[w, z])^(2)-Divide[1,z]*D[w, z]+Divide[\[Alpha]*(w)^(2)+ \[Beta],z]+ \[Gamma]*(w)^(3)+Divide[\[Delta],w]
Failure Failure
Failed [300 / 300]
Result: -3.598076212-.5772156656*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.000000000+2.020860546*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [299 / 300]
Result: Complex[-3.098076211353316, -0.8660254037844389]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.098076211353316, -1.8660254037844388]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E4 d 2 w d z 2 = 1 2 ā¢ w ā¢ ( d w d z ) 2 + 3 2 ā¢ w 3 + 4 ā¢ z ā¢ w 2 + 2 ā¢ ( z 2 - Ī± ) ā¢ w + Ī² w derivative š‘¤ š‘§ 2 1 2 š‘¤ superscript derivative š‘¤ š‘§ 2 3 2 superscript š‘¤ 3 4 š‘§ superscript š‘¤ 2 2 superscript š‘§ 2 š›¼ š‘¤ š›½ š‘¤ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{2w}\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+\frac{3}{2}w^{3}+4zw^{2% }+2(z^{2}-\alpha)w+\frac{\beta}{w}}}
\deriv[2]{w}{z} = \frac{1}{2w}\left(\deriv{w}{z}\right)^{2}+\frac{3}{2}w^{3}+4zw^{2}+2(z^{2}-\alpha)w+\frac{\beta}{w}		 

diff(w, [z$(2)]) = (1)/(2*w)*(diff(w, z))^(2)+(3)/(2)*(w)^(3)+ 4*z*(w)^(2)+ 2*((z)^(2)- alpha)*w +(beta)/(w)

D[w, {z, 2}] == Divide[1,2*w]*(D[w, z])^(2)+Divide[3,2]*(w)^(3)+ 4*z*(w)^(2)+ 2*((z)^(2)- \[Alpha])*w +Divide[\[Beta],w]
Failure Failure
Failed [299 / 300]
Result: 1.299038104-5.250000007*I
Test Values: {alpha = 3/2, beta = 3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: 5.299038110+2.749999997*I
Test Values: {alpha = 3/2, beta = 3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.2990381056766576, -5.25]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5]}


Result: Complex[2.1650635094610964, -5.75]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 0.5]}

... skip entries to safe data
32.2.E5 d 2 w d z 2 = ( 1 2 ā¢ w + 1 w - 1 ) ā¢ ( d w d z ) 2 - 1 z ā¢ d w d z + ( w - 1 ) 2 z 2 ā¢ ( Ī± ā¢ w + Ī² w ) + Ī³ ā¢ w z + Ī“ ā¢ w ā¢ ( w + 1 ) w - 1 derivative š‘¤ š‘§ 2 1 2 š‘¤ 1 š‘¤ 1 superscript derivative š‘¤ š‘§ 2 1 š‘§ derivative š‘¤ š‘§ superscript š‘¤ 1 2 superscript š‘§ 2 š›¼ š‘¤ š›½ š‘¤ š›¾ š‘¤ š‘§ š›æ š‘¤ š‘¤ 1 š‘¤ 1 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\left(% \frac{1}{2w}+\frac{1}{w-1}\right)\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^% {2}-\frac{1}{z}\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{(w-1)^{2}}{z^{2}}\left(% \alpha w+\frac{\beta}{w}\right)+\frac{\gamma w}{z}+\frac{\delta w(w+1)}{w-1}}}
\deriv[2]{w}{z} = \left(\frac{1}{2w}+\frac{1}{w-1}\right)\left(\deriv{w}{z}\right)^{2}-\frac{1}{z}\deriv{w}{z}+\frac{(w-1)^{2}}{z^{2}}\left(\alpha w+\frac{\beta}{w}\right)+\frac{\gamma w}{z}+\frac{\delta w(w+1)}{w-1}		 

diff(w, [z$(2)]) = ((1)/(2*w)+(1)/(w - 1))*(diff(w, z))^(2)-(1)/(z)*diff(w, z)+((w - 1)^(2))/((z)^(2))*(alpha*w +(beta)/(w))+(gamma*w)/(z)+(delta*w*(w + 1))/(w - 1)

D[w, {z, 2}] == (Divide[1,2*w]+Divide[1,w - 1])*(D[w, z])^(2)-Divide[1,z]*D[w, z]+Divide[(w - 1)^(2),(z)^(2)]*(\[Alpha]*w +Divide[\[Beta],w])+Divide[\[Gamma]*w,z]+Divide[\[Delta]*w*(w + 1),w - 1]
Failure Failure
Failed [300 / 300]
Result: -3.206380793+1.517949194*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -3.834936494+2.791317281*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-3.495190528383291, 1.017949192431124]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.1291651245988517, -4.0801270189221945]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E6 d 2 w d z 2 = 1 2 ā¢ ( 1 w + 1 w - 1 + 1 w - z ) ā¢ ( d w d z ) 2 - ( 1 z + 1 z - 1 + 1 w - z ) ā¢ d w d z + w ā¢ ( w - 1 ) ā¢ ( w - z ) z 2 ā¢ ( z - 1 ) 2 ā¢ ( Ī± + Ī² ā¢ z w 2 + Ī³ ā¢ ( z - 1 ) ( w - 1 ) 2 + Ī“ ā¢ z ā¢ ( z - 1 ) ( w - z ) 2 ) derivative š‘¤ š‘§ 2 1 2 1 š‘¤ 1 š‘¤ 1 1 š‘¤ š‘§ superscript derivative š‘¤ š‘§ 2 1 š‘§ 1 š‘§ 1 1 š‘¤ š‘§ derivative š‘¤ š‘§ š‘¤ š‘¤ 1 š‘¤ š‘§ superscript š‘§ 2 superscript š‘§ 1 2 š›¼ š›½ š‘§ superscript š‘¤ 2 š›¾ š‘§ 1 superscript š‘¤ 1 2 š›æ š‘§ š‘§ 1 superscript š‘¤ š‘§ 2 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\frac{\mathrm{d% }w}{\mathrm{d}z}\right)^{2}-\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left% (\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1% )}{(w-z)^{2}}\right)}}
\deriv[2]{w}{z} = \frac{1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\deriv{w}{z}\right)^{2}-\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}\right)\deriv{w}{z}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1)}{(w-z)^{2}}\right)		 

diff(w, [z$(2)]) = (1)/(2)*((1)/(w)+(1)/(w - 1)+(1)/(w - z))*(diff(w, z))^(2)-((1)/(z)+(1)/(z - 1)+(1)/(w - z))*diff(w, z)+(w*(w - 1)*(w - z))/((z)^(2)*(z - 1)^(2))*(alpha +(beta*z)/((w)^(2))+(gamma*(z - 1))/((w - 1)^(2))+(delta*z*(z - 1))/((w - z)^(2)))

D[w, {z, 2}] == Divide[1,2]*(Divide[1,w]+Divide[1,w - 1]+Divide[1,w - z])*(D[w, z])^(2)-(Divide[1,z]+Divide[1,z - 1]+Divide[1,w - z])*D[w, z]+Divide[w*(w - 1)*(w - z),(z)^(2)*(z - 1)^(2)]*(\[Alpha]+Divide[\[Beta]*z,(w)^(2)]+Divide[\[Gamma]*(z - 1),(w - 1)^(2)]+Divide[\[Delta]*z*(z - 1),(w - z)^(2)])
Failure Failure
Failed [269 / 300]
Result: -.9380246356e-1+1.316803425*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: 1.739453154+1.182694224*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2#Ex1 W ā¢ ( Ī¶ ) = a ā¢ ( z ) ā¢ w + b ā¢ ( z ) c ā¢ ( z ) ā¢ w + d ā¢ ( z ) š‘Š šœ š‘Ž š‘§ š‘¤ š‘ š‘§ š‘ š‘§ š‘¤ š‘‘ š‘§ {\displaystyle{\displaystyle W(\zeta)=\frac{a(z)w+b(z)}{c(z)w+d(z)}}}
W(\zeta) = \frac{a(z)w+b(z)}{c(z)w+d(z)}		 

W(zeta) = (a(z)* w + b(z))/(c(z)* w + d(z))
 
W[\[Zeta]] == Divide[a[z]* w + b[z],c[z]* w + d[z]]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex2 Ī¶ = Ļ• ā¢ ( z ) šœ italic-Ļ• š‘§ {\displaystyle{\displaystyle\zeta=\phi(z)}}
\zeta = \phi(z)		 

zeta = phi(z)
 
\[Zeta] == \[Phi][z]
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E9 d 2 u d Ī¶ 2 = 1 u ā¢ ( d u d Ī¶ ) 2 - 1 Ī¶ ā¢ d u d Ī¶ + u 2 ā¢ ( Ī± + Ī³ ā¢ u ) 4 ā¢ Ī¶ 2 + Ī² 4 ā¢ Ī¶ + Ī“ 4 ā¢ u derivative š‘¢ šœ 2 1 š‘¢ superscript derivative š‘¢ šœ 2 1 šœ derivative š‘¢ šœ superscript š‘¢ 2 š›¼ š›¾ š‘¢ 4 superscript šœ 2 š›½ 4 šœ š›æ 4 š‘¢ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=% \frac{1}{u}\left(\frac{\mathrm{d}u}{\mathrm{d}\zeta}\right)^{2}-\frac{1}{\zeta% }\frac{\mathrm{d}u}{\mathrm{d}\zeta}+\frac{u^{2}(\alpha+\gamma u)}{4\zeta^{2}}% +\frac{\beta}{4\zeta}+\frac{\delta}{4u}}}
\deriv[2]{u}{\zeta} = \frac{1}{u}\left(\deriv{u}{\zeta}\right)^{2}-\frac{1}{\zeta}\deriv{u}{\zeta}+\frac{u^{2}(\alpha+\gamma u)}{4\zeta^{2}}+\frac{\beta}{4\zeta}+\frac{\delta}{4u}		 

diff(u, [zeta$(2)]) = (1)/(u)*(diff(u, zeta))^(2)-(1)/(zeta)*diff(u, zeta)+((u)^(2)*(alpha + gamma*u))/(4*(zeta)^(2))+(beta)/(4*zeta)+(delta)/(4*u)

D[u, {\[Zeta], 2}] == Divide[1,u]*(D[u, \[Zeta]])^(2)-Divide[1,\[Zeta]]*D[u, \[Zeta]]+Divide[(u)^(2)*(\[Alpha]+ \[Gamma]*u),4*\[Zeta]^(2)]+Divide[\[Beta],4*\[Zeta]]+Divide[\[Delta],4*u]
Failure Failure
Failed [300 / 300]
Result: -1.074730384+.1153480418*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}


Result: .4374708571+.3969114845*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.0747595264191645, -0.029006350946109677]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4374999999999999, 0.541265877365274]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¶, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E10 d 2 u d z 2 + 1 z ā¢ d u d z = 2 ā¢ Ī± z ā¢ sin ā” u + 2 ā¢ Ī³ ā¢ sin ā” ( 2 ā¢ u ) derivative š‘¢ š‘§ 2 1 š‘§ derivative š‘¢ š‘§ 2 š›¼ š‘§ š‘¢ 2 š›¾ 2 š‘¢ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}z}^{2}}+\frac{% 1}{z}\frac{\mathrm{d}u}{\mathrm{d}z}=\frac{2\alpha}{z}\sin u+2\gamma\sin\left(% 2u\right)}}
\deriv[2]{u}{z}+\frac{1}{z}\deriv{u}{z} = \frac{2\alpha}{z}\sin@@{u}+2\gamma\sin@{2u}		 

diff(u, [z$(2)])+(1)/(z)*diff(u, z) = (2*alpha)/(z)*sin(u)+ 2*gamma*sin(2*u)

D[u, {z, 2}]+Divide[1,z]*D[u, z] == Divide[2*\[Alpha],z]*Sin[u]+ 2*\[Gamma]*Sin[2*u]
Failure Failure
Failed [300 / 300]
Result: -4.496361213+.6291944644*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.346900984+2.955916370*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-5.5647975539874, -0.7848783935570325]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.5418435125289267, -2.4153373252737342]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E11 d 2 u d Ī¶ 2 = 3 ā¢ u 5 + 2 ā¢ Ī¶ ā¢ u 3 + ( 1 4 ā¢ Ī¶ 2 - Ī½ - 1 2 ) ā¢ u + Ī² 32 ā¢ u 3 derivative š‘¢ šœ 2 3 superscript š‘¢ 5 2 šœ superscript š‘¢ 3 1 4 superscript šœ 2 šœˆ 1 2 š‘¢ š›½ 32 superscript š‘¢ 3 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=3u% ^{5}+2\zeta u^{3}+\left(\tfrac{1}{4}\zeta^{2}-\nu-\tfrac{1}{2}\right)u+\frac{% \beta}{32u^{3}}}}
\deriv[2]{u}{\zeta} = 3u^{5}+2\zeta u^{3}+\left(\tfrac{1}{4}\zeta^{2}-\nu-\tfrac{1}{2}\right)u+\frac{\beta}{32u^{3}}		 

diff(u, [zeta$(2)]) = 3*(u)^(5)+ 2*zeta*(u)^(3)+((1)/(4)*(zeta)^(2)- nu -(1)/(2))*u +(beta)/(32*(u)^(3))

D[u, {\[Zeta], 2}] == 3*(u)^(5)+ 2*\[Zeta]*(u)^(3)+(Divide[1,4]*\[Zeta]^(2)- \[Nu]-Divide[1,2])*u +Divide[\[Beta],32*(u)^(3)]
Failure Failure
Failed [300 / 300]
Result: 4.531088915-2.319150408*I
Test Values: {beta = 3/2, nu = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}


Result: 5.263139725+.9129004010*I
Test Values: {beta = 3/2, nu = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[4.531088913245536, -2.3191504037844384]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī², 1.5], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.165063509461097, -2.685175807568877]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī², 1.5], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E12 d 2 u d Ī¶ 2 = - Ī± ā¢ cosh ā” u 2 ā¢ ( sinh ā” u ) 3 - Ī² ā¢ sinh ā” u 2 ā¢ ( cosh ā” u ) 3 - 1 4 ā¢ Ī³ ā¢ e Ī¶ ā¢ sinh ā” ( 2 ā¢ u ) - 1 8 ā¢ Ī“ ā¢ e 2 ā¢ Ī¶ ā¢ sinh ā” ( 4 ā¢ u ) derivative š‘¢ šœ 2 š›¼ š‘¢ 2 superscript š‘¢ 3 š›½ š‘¢ 2 superscript š‘¢ 3 1 4 š›¾ superscript š‘’ šœ 2 š‘¢ 1 8 š›æ superscript š‘’ 2 šœ 4 š‘¢ {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}u}{{\mathrm{d}\zeta}^{2}}=-% \frac{\alpha\cosh u}{2(\sinh u)^{3}}-\frac{\beta\sinh u}{2(\cosh u)^{3}}-% \tfrac{1}{4}\gamma e^{\zeta}\sinh\left(2u\right)-\tfrac{1}{8}\delta e^{2\zeta}% \sinh\left(4u\right)}}
\deriv[2]{u}{\zeta} = -\frac{\alpha\cosh@@{u}}{2(\sinh@@{u})^{3}}-\frac{\beta\sinh@@{u}}{2(\cosh@@{u})^{3}}-\tfrac{1}{4}\gamma e^{\zeta}\sinh@{2u}-\tfrac{1}{8}\delta e^{2\zeta}\sinh@{4u}		 

diff(u, [zeta$(2)]) = -(alpha*cosh(u))/(2*(sinh(u))^(3))-(beta*sinh(u))/(2*(cosh(u))^(3))-(1)/(4)*gamma*exp(zeta)*sinh(2*u)-(1)/(8)*delta*exp(2*zeta)*sinh(4*u)

D[u, {\[Zeta], 2}] == -Divide[\[Alpha]*Cosh[u],2*(Sinh[u])^(3)]-Divide[\[Beta]*Sinh[u],2*(Cosh[u])^(3)]-Divide[1,4]*\[Gamma]*Exp[\[Zeta]]*Sinh[2*u]-Divide[1,8]*\[Delta]*Exp[2*\[Zeta]]*Sinh[4*u]
Failure Failure
Failed [300 / 300]
Result: -10.15437375-4.132059394*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}


Result: -.1182986371-1.333346640*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-10.983749451492802, -3.604532198424999]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3640397236276506, -1.2834088930332135]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī±, 1.5], Rule[Ī², 1.5], Rule[Ī³, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī“, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¶, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E13 z ā¢ ( 1 - z ) ā¢ I ā¢ ( āˆ« āˆž w d t t ā¢ ( t - 1 ) ā¢ ( t - z ) ) = w ā¢ ( w - 1 ) ā¢ ( w - z ) ā¢ ( Ī± + Ī² ā¢ z w 2 + Ī³ ā¢ ( z - 1 ) ( w - 1 ) 2 + ( Ī“ - 1 2 ) ā¢ z ā¢ ( z - 1 ) ( w - z ) 2 ) š‘§ 1 š‘§ š¼ superscript subscript š‘¤ š‘” š‘” š‘” 1 š‘” š‘§ š‘¤ š‘¤ 1 š‘¤ š‘§ š›¼ š›½ š‘§ superscript š‘¤ 2 š›¾ š‘§ 1 superscript š‘¤ 1 2 š›æ 1 2 š‘§ š‘§ 1 superscript š‘¤ š‘§ 2 {\displaystyle{\displaystyle z(1-z)I\left(\int_{\infty}^{w}\frac{\mathrm{d}t}{% \sqrt{t(t-1)(t-z)}}\right)=\sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{% 2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}% }\right)}}
z(1-z)I\left(\int_{\infty}^{w}\frac{\diff{t}}{\sqrt{t(t-1)(t-z)}}\right) = \sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}}\right)		 

z*(1 - z)*I(int((1)/(sqrt(t*(t - 1)*(t - z))), t = infinity..w)) = sqrt(w*(w - 1)*(w - z))*(alpha +(beta*z)/((w)^(2))+(gamma*(z - 1))/((w - 1)^(2))+(delta -(1)/(2))*(z*(z - 1))/((w - z)^(2)))

z*(1 - z)*I[(Integrate[Divide[1,Sqrt[t*(t - 1)*(t - z)]], {t, Infinity, w}, GenerateConditions->None]) ] == Sqrt[w*(w - 1)*(w - z)]*(\[Alpha]+Divide[\[Beta]*z,(w)^(2)]+Divide[\[Gamma]*(z - 1),(w - 1)^(2)]+(\[Delta]-Divide[1,2])*Divide[z*(z - 1),(w - z)^(2)])
Failure Aborted Skipped - Because timed out Skipped - Because timed out
32.2.E14 I = z ā¢ ( 1 - z ) ā¢ d 2 d z 2 + ( 1 - 2 ā¢ z ) ā¢ d d z - 1 4 š¼ š‘§ 1 š‘§ derivative š‘§ 2 1 2 š‘§ derivative š‘§ 1 4 {\displaystyle{\displaystyle I=z(1-z)\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}% }+(1-2z)\frac{\mathrm{d}}{\mathrm{d}z}-\frac{1}{4}}}
I = z(1-z)\deriv[2]{}{z}+(1-2z)\deriv{}{z}-\frac{1}{4}		 

I = z*(1 - z)*diff(+(1 - 2*z)*diff(-, z), [z$(2)])(1)/(4)

I == z*(1 - z)*D[+(1 - 2*z)*D[-, z], {z, 2}]Divide[1,4]
Error Failure - Error
32.2#Ex3 d f 1 d z + f 1 ā¢ ( f 2 - f 3 ) + 2 ā¢ Ī¼ 1 = 0 derivative subscript š‘“ 1 š‘§ subscript š‘“ 1 subscript š‘“ 2 subscript š‘“ 3 2 subscript šœ‡ 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{1}}{\mathrm{d}z}+f_{1}(f_{2}-f_% {3})+2\mu_{1}=0}}
\deriv{f_{1}}{z}+f_{1}(f_{2}-f_{3})+2\mu_{1} = 0		 

diff(f[1], z)+ f[1]*(f[2]- f[3])+ 2*mu[1] = 0

D[Subscript[f, 1], z]+ Subscript[f, 1]*(Subscript[f, 2]- Subscript[f, 3])+ 2*Subscript[\[Mu], 1] == 0
Failure Failure
Failed [300 / 300]
Result: 1.732050808+1.000000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I}


Result: -1.000000000+1.732050808*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.7320508075688774, 0.9999999999999999]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9999999999999996, 1.7320508075688774]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2#Ex4 d f 2 d z + f 2 ā¢ ( f 3 - f 1 ) + 2 ā¢ Ī¼ 2 = 0 derivative subscript š‘“ 2 š‘§ subscript š‘“ 2 subscript š‘“ 3 subscript š‘“ 1 2 subscript šœ‡ 2 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{2}}{\mathrm{d}z}+f_{2}(f_{3}-f_% {1})+2\mu_{2}=0}}
\deriv{f_{2}}{z}+f_{2}(f_{3}-f_{1})+2\mu_{2} = 0		 

diff(f[2], z)+ f[2]*(f[3]- f[1])+ 2*mu[2] = 0

D[Subscript[f, 2], z]+ Subscript[f, 2]*(Subscript[f, 3]- Subscript[f, 1])+ 2*Subscript[\[Mu], 2] == 0
Failure Failure
Failed [300 / 300]
Result: 1.732050808+1.000000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I}


Result: -1.000000000+1.732050808*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.7320508075688774, 0.9999999999999999]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9999999999999996, 1.7320508075688774]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2#Ex5 d f 3 d z + f 3 ā¢ ( f 1 - f 2 ) + 2 ā¢ Ī¼ 3 = 0 derivative subscript š‘“ 3 š‘§ subscript š‘“ 3 subscript š‘“ 1 subscript š‘“ 2 2 subscript šœ‡ 3 0 {\displaystyle{\displaystyle\frac{\mathrm{d}f_{3}}{\mathrm{d}z}+f_{3}(f_{1}-f_% {2})+2\mu_{3}=0}}
\deriv{f_{3}}{z}+f_{3}(f_{1}-f_{2})+2\mu_{3} = 0		 

diff(f[3], z)+ f[3]*(f[1]- f[2])+ 2*mu[3] = 0

D[Subscript[f, 3], z]+ Subscript[f, 3]*(Subscript[f, 1]- Subscript[f, 2])+ 2*Subscript[\[Mu], 3] == 0
Failure Failure
Failed [300 / 300]
Result: 1.732050808+1.000000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[3] = 1/2*3^(1/2)+1/2*I}


Result: -1.000000000+1.732050808*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, mu[3] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.7320508075688774, 0.9999999999999999]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.9999999999999996, 1.7320508075688774]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ī¼, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[Ī¼, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.2.E16 Ī¼ 1 + Ī¼ 2 + Ī¼ 3 = 1 subscript šœ‡ 1 subscript šœ‡ 2 subscript šœ‡ 3 1 {\displaystyle{\displaystyle\mu_{1}+\mu_{2}+\mu_{3}=1}}
\mu_{1}+\mu_{2}+\mu_{3} = 1		 

mu[1]+ mu[2]+ mu[3] = 1
 
Subscript[\[Mu], 1]+ Subscript[\[Mu], 2]+ Subscript[\[Mu], 3] == 1
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E17 f 1 ā¢ ( z ) + f 2 ā¢ ( z ) + f 3 ā¢ ( z ) + 2 ā¢ z = 0 subscript š‘“ 1 š‘§ subscript š‘“ 2 š‘§ subscript š‘“ 3 š‘§ 2 š‘§ 0 {\displaystyle{\displaystyle f_{1}(z)+f_{2}(z)+f_{3}(z)+2z=0}}
f_{1}(z)+f_{2}(z)+f_{3}(z)+2z = 0		 

f[1](z)+ f[2](z)+ f[3](z)+ 2*z = 0
 
Subscript[f, 1][z]+ Subscript[f, 2][z]+ Subscript[f, 3][z]+ 2*z == 0
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E18 ( Ī± , Ī² ) = ( Ī¼ 3 - Ī¼ 2 , - 2 ā¢ Ī¼ 1 2 ) š›¼ š›½ subscript šœ‡ 3 subscript šœ‡ 2 2 superscript subscript šœ‡ 1 2 {\displaystyle{\displaystyle(\alpha,\beta)=(\mu_{3}-\mu_{2},-2\mu_{1}^{2})}}
(\alpha,\beta) = (\mu_{3}-\mu_{2},-2\mu_{1}^{2})		 

(alpha , beta) = (mu[3]- mu[2], - 2*(mu[1])^(2))
 
(\[Alpha], \[Beta]) == (Subscript[\[Mu], 3]- Subscript[\[Mu], 2], - 2*(Subscript[\[Mu], 1])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex6 z ā¢ d f 1 d z = f 1 ā¢ f 3 ā¢ ( f 2 - f 4 ) + ( 1 2 - Ī¼ 3 ) ā¢ f 1 + Ī¼ 1 ā¢ f 3 š‘§ derivative subscript š‘“ 1 š‘§ subscript š‘“ 1 subscript š‘“ 3 subscript š‘“ 2 subscript š‘“ 4 1 2 subscript šœ‡ 3 subscript š‘“ 1 subscript šœ‡ 1 subscript š‘“ 3 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{1}}{\mathrm{d}z}=f_{1}f_{3}(f% _{2}-f_{4})+(\tfrac{1}{2}-\mu_{3})f_{1}+\mu_{1}f_{3}}}
z\deriv{f_{1}}{z} = f_{1}f_{3}(f_{2}-f_{4})+(\tfrac{1}{2}-\mu_{3})f_{1}+\mu_{1}f_{3}		 

z*diff(f[1], z) = f[1]*f[3]*(f[2]- f[4])+((1)/(2)- mu[3])*f[1]+ mu[1]*f[3]
 
z*D[Subscript[f, 1], z] == Subscript[f, 1]*Subscript[f, 3]*(Subscript[f, 2]- Subscript[f, 4])+(Divide[1,2]- Subscript[\[Mu], 3])*Subscript[f, 1]+ Subscript[\[Mu], 1]*Subscript[f, 3]
 Failure Failure
Failed [298 / 300]
Result: -.4330127020-.2500000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[3] = 1/2*3^(1/2)+1/2*I}

Result: -1.799038106-.6160254036*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[3] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
32.2#Ex7 z ā¢ d f 2 d z = f 2 ā¢ f 4 ā¢ ( f 3 - f 1 ) + ( 1 2 - Ī¼ 4 ) ā¢ f 2 + Ī¼ 2 ā¢ f 4 š‘§ derivative subscript š‘“ 2 š‘§ subscript š‘“ 2 subscript š‘“ 4 subscript š‘“ 3 subscript š‘“ 1 1 2 subscript šœ‡ 4 subscript š‘“ 2 subscript šœ‡ 2 subscript š‘“ 4 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{2}}{\mathrm{d}z}=f_{2}f_{4}(f% _{3}-f_{1})+(\tfrac{1}{2}-\mu_{4})f_{2}+\mu_{2}f_{4}}}
z\deriv{f_{2}}{z} = f_{2}f_{4}(f_{3}-f_{1})+(\tfrac{1}{2}-\mu_{4})f_{2}+\mu_{2}f_{4}		 

z*diff(f[2], z) = f[2]*f[4]*(f[3]- f[1])+((1)/(2)- mu[4])*f[2]+ mu[2]*f[4]
 
z*D[Subscript[f, 2], z] == Subscript[f, 2]*Subscript[f, 4]*(Subscript[f, 3]- Subscript[f, 1])+(Divide[1,2]- Subscript[\[Mu], 4])*Subscript[f, 2]+ Subscript[\[Mu], 2]*Subscript[f, 4]
 Failure Failure
Failed [298 / 300]
Result: -.4330127020-.2500000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I, mu[4] = 1/2*3^(1/2)+1/2*I}

Result: -1.799038106-.6160254036*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I, mu[4] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
32.2#Ex8 z ā¢ d f 3 d z = f 3 ā¢ f 1 ā¢ ( f 4 - f 2 ) + ( 1 2 - Ī¼ 1 ) ā¢ f 3 + Ī¼ 3 ā¢ f 1 š‘§ derivative subscript š‘“ 3 š‘§ subscript š‘“ 3 subscript š‘“ 1 subscript š‘“ 4 subscript š‘“ 2 1 2 subscript šœ‡ 1 subscript š‘“ 3 subscript šœ‡ 3 subscript š‘“ 1 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{3}}{\mathrm{d}z}=f_{3}f_{1}(f% _{4}-f_{2})+(\tfrac{1}{2}-\mu_{1})f_{3}+\mu_{3}f_{1}}}
z\deriv{f_{3}}{z} = f_{3}f_{1}(f_{4}-f_{2})+(\tfrac{1}{2}-\mu_{1})f_{3}+\mu_{3}f_{1}		 

z*diff(f[3], z) = f[3]*f[1]*(f[4]- f[2])+((1)/(2)- mu[1])*f[3]+ mu[3]*f[1]
 
z*D[Subscript[f, 3], z] == Subscript[f, 3]*Subscript[f, 1]*(Subscript[f, 4]- Subscript[f, 2])+(Divide[1,2]- Subscript[\[Mu], 1])*Subscript[f, 3]+ Subscript[\[Mu], 3]*Subscript[f, 1]
 Failure Failure
Failed [298 / 300]
Result: -.4330127020-.2500000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[3] = 1/2*3^(1/2)+1/2*I}

Result: .9330127024+.1160254036*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[1] = 1/2*3^(1/2)+1/2*I, mu[3] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
32.2#Ex9 z ā¢ d f 4 d z = f 4 ā¢ f 2 ā¢ ( f 1 - f 3 ) + ( 1 2 - Ī¼ 2 ) ā¢ f 4 + Ī¼ 4 ā¢ f 2 š‘§ derivative subscript š‘“ 4 š‘§ subscript š‘“ 4 subscript š‘“ 2 subscript š‘“ 1 subscript š‘“ 3 1 2 subscript šœ‡ 2 subscript š‘“ 4 subscript šœ‡ 4 subscript š‘“ 2 {\displaystyle{\displaystyle z\frac{\mathrm{d}f_{4}}{\mathrm{d}z}=f_{4}f_{2}(f% _{1}-f_{3})+(\tfrac{1}{2}-\mu_{2})f_{4}+\mu_{4}f_{2}}}
z\deriv{f_{4}}{z} = f_{4}f_{2}(f_{1}-f_{3})+(\tfrac{1}{2}-\mu_{2})f_{4}+\mu_{4}f_{2}		 

z*diff(f[4], z) = f[4]*f[2]*(f[1]- f[3])+((1)/(2)- mu[2])*f[4]+ mu[4]*f[2]
 
z*D[Subscript[f, 4], z] == Subscript[f, 4]*Subscript[f, 2]*(Subscript[f, 1]- Subscript[f, 3])+(Divide[1,2]- Subscript[\[Mu], 2])*Subscript[f, 4]+ Subscript[\[Mu], 4]*Subscript[f, 2]
 Failure Failure
Failed [298 / 300]
Result: -.4330127020-.2500000000*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I, mu[4] = 1/2*3^(1/2)+1/2*I}

Result: .9330127024+.1160254036*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, f[1] = 1/2*3^(1/2)+1/2*I, f[2] = 1/2*3^(1/2)+1/2*I, f[3] = 1/2*3^(1/2)+1/2*I, f[4] = 1/2*3^(1/2)+1/2*I, mu[2] = 1/2*3^(1/2)+1/2*I, mu[4] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
32.2.E20 Ī¼ 1 + Ī¼ 2 + Ī¼ 3 + Ī¼ 4 = 1 subscript šœ‡ 1 subscript šœ‡ 2 subscript šœ‡ 3 subscript šœ‡ 4 1 {\displaystyle{\displaystyle\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}=1}}
\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4} = 1		 

mu[1]+ mu[2]+ mu[3]+ mu[4] = 1
 
Subscript[\[Mu], 1]+ Subscript[\[Mu], 2]+ Subscript[\[Mu], 3]+ Subscript[\[Mu], 4] == 1
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E21 f 1 ā¢ ( z ) + f 3 ā¢ ( z ) = z subscript š‘“ 1 š‘§ subscript š‘“ 3 š‘§ š‘§ {\displaystyle{\displaystyle f_{1}(z)+f_{3}(z)=\sqrt{z}}}
f_{1}(z)+f_{3}(z) = \sqrt{z}		 

f[1](z)+ f[3](z) = sqrt(z)
 
Subscript[f, 1][z]+ Subscript[f, 3][z] == Sqrt[z]
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E22 f 2 ā¢ ( z ) + f 4 ā¢ ( z ) = z subscript š‘“ 2 š‘§ subscript š‘“ 4 š‘§ š‘§ {\displaystyle{\displaystyle f_{2}(z)+f_{4}(z)=\sqrt{z}}}
f_{2}(z)+f_{4}(z) = \sqrt{z}		 

f[2](z)+ f[4](z) = sqrt(z)
 
Subscript[f, 2][z]+ Subscript[f, 4][z] == Sqrt[z]
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E23 ( Ī± , Ī² , Ī³ , Ī“ ) = ( 1 2 ā¢ Ī¼ 1 2 , - 1 2 ā¢ Ī¼ 3 2 , Ī¼ 4 - Ī¼ 2 , - 1 2 ) š›¼ š›½ š›¾ š›æ 1 2 superscript subscript šœ‡ 1 2 1 2 superscript subscript šœ‡ 3 2 subscript šœ‡ 4 subscript šœ‡ 2 1 2 {\displaystyle{\displaystyle(\alpha,\beta,\gamma,\delta)=(\tfrac{1}{2}\mu_{1}^% {2},-\tfrac{1}{2}\mu_{3}^{2},\mu_{4}-\mu_{2},-\tfrac{1}{2})}}
(\alpha,\beta,\gamma,\delta) = (\tfrac{1}{2}\mu_{1}^{2},-\tfrac{1}{2}\mu_{3}^{2},\mu_{4}-\mu_{2},-\tfrac{1}{2})		 

(alpha , beta , gamma , delta) = ((1)/(2)*(mu[1])^(2), -(1)/(2)*(mu[3])^(2), mu[4]- mu[2], -(1)/(2))
 
(\[Alpha], \[Beta], \[Gamma], \[Delta]) == (Divide[1,2]*(Subscript[\[Mu], 1])^(2), -Divide[1,2]*(Subscript[\[Mu], 3])^(2), Subscript[\[Mu], 4]- Subscript[\[Mu], 2], -Divide[1,2])
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E25 w ā¢ ( z ; Ī± ) = Ļµ ā¢ W ā¢ ( Ī¶ ) + 1 Ļµ 5 š‘¤ š‘§ š›¼ italic-Ļµ š‘Š šœ 1 superscript italic-Ļµ 5 {\displaystyle{\displaystyle w(z;\alpha)=\epsilon W(\zeta)+\frac{1}{\epsilon^{% 5}}}}
w(z;\alpha) = \epsilon W(\zeta)+\frac{1}{\epsilon^{5}}		 

w(z ; alpha) = epsilon*W(zeta)+(1)/((epsilon)^(5))
 
w[z ; \[Alpha]] == \[Epsilon]*W[\[Zeta]]+Divide[1,\[Epsilon]^(5)]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex10 z = Ļµ 2 ā¢ Ī¶ - 6 Ļµ 10 š‘§ superscript italic-Ļµ 2 šœ 6 superscript italic-Ļµ 10 {\displaystyle{\displaystyle z=\epsilon^{2}\zeta-\frac{6}{\epsilon^{10}}}}
z = \epsilon^{2}\zeta-\frac{6}{\epsilon^{10}}		 

z = (epsilon)^(2)* zeta -(6)/((epsilon)^(10))
 
z == \[Epsilon]^(2)* \[Zeta]-Divide[6,\[Epsilon]^(10)]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex11 Ī± = 4 Ļµ 15 š›¼ 4 superscript italic-Ļµ 15 {\displaystyle{\displaystyle\alpha=\frac{4}{\epsilon^{15}}}}
\alpha = \frac{4}{\epsilon^{15}}		 

alpha = (4)/((epsilon)^(15))
 
\[Alpha] == Divide[4,\[Epsilon]^(15)]
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E27 d 2 W d Ī¶ 2 = 6 ā¢ W 2 + Ī¶ + Ļµ 6 ā¢ ( 2 ā¢ W 3 + Ī¶ ā¢ W ) derivative š‘Š šœ 2 6 superscript š‘Š 2 šœ superscript italic-Ļµ 6 2 superscript š‘Š 3 šœ š‘Š {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\zeta}^{2}}=6W% ^{2}+\zeta+\epsilon^{6}(2W^{3}+\zeta W)}}
\deriv[2]{W}{\zeta} = 6W^{2}+\zeta+\epsilon^{6}(2W^{3}+\zeta W)		 

diff(W, [zeta$(2)]) = 6*(W)^(2)+ zeta + (epsilon)^(6)*(2*(W)^(3)+ zeta*W)
 
D[W, {\[Zeta], 2}] == 6*(W)^(2)+ \[Zeta]+ \[Epsilon]^(6)*(2*(W)^(3)+ \[Zeta]*W)
 Failure Failure
Failed [300 / 300]
Result: -4.366025408-8.562177830*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, epsilon = 1}

Result: -35.86602547-189.1217784*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, epsilon = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-4.366025403784439, -8.56217782649107]
Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ļµ, 1], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-35.86602540378445, -189.1217782649107]
Test Values: {Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ļµ, 2], Rule[Ī¶, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.2.E28 w ā¢ ( z ; Ī± , Ī² , Ī³ , Ī“ ) = 1 + 2 ā¢ Ļµ ā¢ W ā¢ ( Ī¶ ; a ) š‘¤ š‘§ š›¼ š›½ š›¾ š›æ 1 2 italic-Ļµ š‘Š šœ š‘Ž {\displaystyle{\displaystyle w(z;\alpha,\beta,\gamma,\delta)=1+2\epsilon W(% \zeta;a)}}
w(z;\alpha,\beta,\gamma,\delta) = 1+2\epsilon W(\zeta;a)		 

w(z ; alpha , beta , gamma , delta) = 1 + 2*epsilon*W(zeta ; a)
 
w[z ; \[Alpha], \[Beta], \[Gamma], \[Delta]] == 1 + 2*\[Epsilon]*W[\[Zeta]; a]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex12 z = 1 + Ļµ 2 ā¢ Ī¶ š‘§ 1 superscript italic-Ļµ 2 šœ {\displaystyle{\displaystyle z=1+\epsilon^{2}\zeta}}
z = 1+\epsilon^{2}\zeta		 

z = 1 + (epsilon)^(2)* zeta
 
z == 1 + \[Epsilon]^(2)* \[Zeta]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex13 Ī± = - 1 2 ā¢ Ļµ - 6 š›¼ 1 2 superscript italic-Ļµ 6 {\displaystyle{\displaystyle\alpha=-\tfrac{1}{2}\epsilon^{-6}}}
\alpha = -\tfrac{1}{2}\epsilon^{-6}		 

alpha = -(1)/(2)*(epsilon)^(- 6)
 
\[Alpha] == -Divide[1,2]*\[Epsilon]^(- 6)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex14 Ī² = 1 2 ā¢ Ļµ - 6 + 2 ā¢ a ā¢ Ļµ - 3 š›½ 1 2 superscript italic-Ļµ 6 2 š‘Ž superscript italic-Ļµ 3 {\displaystyle{\displaystyle\beta=\tfrac{1}{2}\epsilon^{-6}+2a\epsilon^{-3}}}
\beta = \tfrac{1}{2}\epsilon^{-6}+2a\epsilon^{-3}		 

beta = (1)/(2)*(epsilon)^(- 6)+ 2*a*(epsilon)^(- 3)
 
\[Beta] == Divide[1,2]*\[Epsilon]^(- 6)+ 2*a*\[Epsilon]^(- 3)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex15 Ī³ = - Ī“ š›¾ š›æ {\displaystyle{\displaystyle\gamma=-\delta}}
\gamma = -\delta		 

gamma = - delta
 
\[Gamma] == - \[Delta]
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E30 w ā¢ ( z ; Ī± , Ī² ) = 2 2 / 3 ā¢ Ļµ - 1 ā¢ W ā¢ ( Ī¶ ; a ) + Ļµ - 3 š‘¤ š‘§ š›¼ š›½ superscript 2 2 3 superscript italic-Ļµ 1 š‘Š šœ š‘Ž superscript italic-Ļµ 3 {\displaystyle{\displaystyle w(z;\alpha,\beta)=2^{2/3}\epsilon^{-1}W(\zeta;a)+% \epsilon^{-3}}}
w(z;\alpha,\beta) = 2^{2/3}\epsilon^{-1}W(\zeta;a)+\epsilon^{-3}		 

w(z ; alpha , beta) = (2)^(2/3)* (epsilon)^(- 1)* W(zeta ; a)+ (epsilon)^(- 3)
 
w[z ; \[Alpha], \[Beta]] == (2)^(2/3)* \[Epsilon]^(- 1)* W[\[Zeta]; a]+ \[Epsilon]^(- 3)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex16 z = 2 - 2 / 3 ā¢ Ļµ ā¢ Ī¶ - Ļµ - 3 š‘§ superscript 2 2 3 italic-Ļµ šœ superscript italic-Ļµ 3 {\displaystyle{\displaystyle z=2^{-2/3}\epsilon\zeta-\epsilon^{-3}}}
z = 2^{-2/3}\epsilon\zeta-\epsilon^{-3}		 

z = (2)^(- 2/3)* epsilon*zeta - (epsilon)^(- 3)
 
z == (2)^(- 2/3)* \[Epsilon]*\[Zeta]- \[Epsilon]^(- 3)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex17 Ī± = - 2 ā¢ a - 1 2 ā¢ Ļµ - 6 š›¼ 2 š‘Ž 1 2 superscript italic-Ļµ 6 {\displaystyle{\displaystyle\alpha=-2a-\tfrac{1}{2}\epsilon^{-6}}}
\alpha = -2a-\tfrac{1}{2}\epsilon^{-6}		 

alpha = - 2*a -(1)/(2)*(epsilon)^(- 6)
 
\[Alpha] == - 2*a -Divide[1,2]*\[Epsilon]^(- 6)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex18 Ī² = - 1 2 ā¢ Ļµ - 12 š›½ 1 2 superscript italic-Ļµ 12 {\displaystyle{\displaystyle\beta=-\tfrac{1}{2}\epsilon^{-12}}}
\beta = -\tfrac{1}{2}\epsilon^{-12}		 

beta = -(1)/(2)*(epsilon)^(- 12)
 
\[Beta] == -Divide[1,2]*\[Epsilon]^(- 12)
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E32 w ā¢ ( z ; Ī± , Ī² , Ī³ , Ī“ ) = 1 + Ļµ ā¢ Ī¶ ā¢ W ā¢ ( Ī¶ ; a , b , c , d ) š‘¤ š‘§ š›¼ š›½ š›¾ š›æ 1 italic-Ļµ šœ š‘Š šœ š‘Ž š‘ š‘ š‘‘ {\displaystyle{\displaystyle w(z;\alpha,\beta,\gamma,\delta)=1+\epsilon\zeta W% (\zeta;a,b,c,d)}}
w(z;\alpha,\beta,\gamma,\delta) = 1+\epsilon\zeta W(\zeta;a,b,c,d)		 

w(z ; alpha , beta , gamma , delta) = 1 + epsilon*zeta*W(zeta ; a , b , c , d)
 
w[z ; \[Alpha], \[Beta], \[Gamma], \[Delta]] == 1 + \[Epsilon]*\[Zeta]*W[\[Zeta]; a , b , c , d]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex19 z = Ī¶ 2 š‘§ superscript šœ 2 {\displaystyle{\displaystyle z=\zeta^{2}}}
z = \zeta^{2}		 

z = (zeta)^(2)
 
z == \[Zeta]^(2)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex20 Ī± = 1 4 ā¢ a ā¢ Ļµ - 1 + 1 8 ā¢ c ā¢ Ļµ - 2 š›¼ 1 4 š‘Ž superscript italic-Ļµ 1 1 8 š‘ superscript italic-Ļµ 2 {\displaystyle{\displaystyle\alpha=\tfrac{1}{4}a\epsilon^{-1}+\tfrac{1}{8}c% \epsilon^{-2}}}
\alpha = \tfrac{1}{4}a\epsilon^{-1}+\tfrac{1}{8}c\epsilon^{-2}		 

alpha = (1)/(4)*a*(epsilon)^(- 1)+(1)/(8)*c*(epsilon)^(- 2)
 
\[Alpha] == Divide[1,4]*a*\[Epsilon]^(- 1)+Divide[1,8]*c*\[Epsilon]^(- 2)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex21 Ī² = - 1 8 ā¢ c ā¢ Ļµ - 2 š›½ 1 8 š‘ superscript italic-Ļµ 2 {\displaystyle{\displaystyle\beta=-\tfrac{1}{8}c\epsilon^{-2}}}
\beta = -\tfrac{1}{8}c\epsilon^{-2}		 

beta = -(1)/(8)*c*(epsilon)^(- 2)
 
\[Beta] == -Divide[1,8]*c*\[Epsilon]^(- 2)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex22 Ī³ = 1 4 ā¢ Ļµ ā¢ b š›¾ 1 4 italic-Ļµ š‘ {\displaystyle{\displaystyle\gamma=\tfrac{1}{4}\epsilon b}}
\gamma = \tfrac{1}{4}\epsilon b		 

gamma = (1)/(4)*epsilon*b
 
\[Gamma] == Divide[1,4]*\[Epsilon]*b
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex23 Ī“ = 1 8 ā¢ Ļµ 2 ā¢ d š›æ 1 8 superscript italic-Ļµ 2 š‘‘ {\displaystyle{\displaystyle\delta=\tfrac{1}{8}\epsilon^{2}d}}
\delta = \tfrac{1}{8}\epsilon^{2}d		 

delta = (1)/(8)*(epsilon)^(2)* d
 
\[Delta] == Divide[1,8]*\[Epsilon]^(2)* d
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E34 w ā¢ ( z ; Ī± , Ī² , Ī³ , Ī“ ) = 1 2 ā¢ 2 ā¢ Ļµ ā¢ W ā¢ ( Ī¶ ; a , b ) š‘¤ š‘§ š›¼ š›½ š›¾ š›æ 1 2 2 italic-Ļµ š‘Š šœ š‘Ž š‘ {\displaystyle{\displaystyle w(z;\alpha,\beta,\gamma,\delta)=\tfrac{1}{2}\sqrt% {2}\epsilon W(\zeta;a,b)}}
w(z;\alpha,\beta,\gamma,\delta) = \tfrac{1}{2}\sqrt{2}\epsilon W(\zeta;a,b)		 

w(z ; alpha , beta , gamma , delta) = (1)/(2)*sqrt(2)*epsilon*W(zeta ; a , b)
 
w[z ; \[Alpha], \[Beta], \[Gamma], \[Delta]] == Divide[1,2]*Sqrt[2]*\[Epsilon]*W[\[Zeta]; a , b]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex24 z = 1 + 2 ā¢ Ļµ ā¢ Ī¶ š‘§ 1 2 italic-Ļµ šœ {\displaystyle{\displaystyle z=1+\sqrt{2}\epsilon\zeta}}
z = 1+\sqrt{2}\epsilon\zeta		 

z = 1 +sqrt(2)*epsilon*zeta
 
z == 1 +Sqrt[2]*\[Epsilon]*\[Zeta]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex25 Ī± = 1 2 ā¢ Ļµ - 4 š›¼ 1 2 superscript italic-Ļµ 4 {\displaystyle{\displaystyle\alpha=\tfrac{1}{2}\epsilon^{-4}}}
\alpha = \tfrac{1}{2}\epsilon^{-4}		 

alpha = (1)/(2)*(epsilon)^(- 4)
 
\[Alpha] == Divide[1,2]*\[Epsilon]^(- 4)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex26 Ī² = 1 4 ā¢ b š›½ 1 4 š‘ {\displaystyle{\displaystyle\beta=\tfrac{1}{4}b}}
\beta = \tfrac{1}{4}b		 

beta = (1)/(4)*b
 
\[Beta] == Divide[1,4]*b
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex27 Ī³ = - Ļµ - 4 š›¾ superscript italic-Ļµ 4 {\displaystyle{\displaystyle\gamma=-\epsilon^{-4}}}
\gamma = -\epsilon^{-4}		 

gamma = - (epsilon)^(- 4)
 
\[Gamma] == - \[Epsilon]^(- 4)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex28 Ī“ = a ā¢ Ļµ - 2 - 1 2 ā¢ Ļµ - 4 š›æ š‘Ž superscript italic-Ļµ 2 1 2 superscript italic-Ļµ 4 {\displaystyle{\displaystyle\delta=a\epsilon^{-2}-\tfrac{1}{2}\epsilon^{-4}}}
\delta = a\epsilon^{-2}-\tfrac{1}{2}\epsilon^{-4}		 

delta = a*(epsilon)^(- 2)-(1)/(2)*(epsilon)^(- 4)
 
\[Delta] == a*\[Epsilon]^(- 2)-Divide[1,2]*\[Epsilon]^(- 4)
 Skipped - no semantic math Skipped - no semantic math - -
32.2.E36 w ā¢ ( z ; Ī± , Ī² , Ī³ , Ī“ ) = W ā¢ ( Ī¶ ; a , b , c , d ) š‘¤ š‘§ š›¼ š›½ š›¾ š›æ š‘Š šœ š‘Ž š‘ š‘ š‘‘ {\displaystyle{\displaystyle w(z;\alpha,\beta,\gamma,\delta)=W(\zeta;a,b,c,d)}}
w(z;\alpha,\beta,\gamma,\delta) = W(\zeta;a,b,c,d)		 

w(z ; alpha , beta , gamma , delta) = W(zeta ; a , b , c , d)
 
w[z ; \[Alpha], \[Beta], \[Gamma], \[Delta]] == W[\[Zeta]; a , b , c , d]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex29 z = 1 + Ļµ ā¢ Ī¶ š‘§ 1 italic-Ļµ šœ {\displaystyle{\displaystyle z=1+\epsilon\zeta}}
z = 1+\epsilon\zeta		 

z = 1 + epsilon*zeta
 
z == 1 + \[Epsilon]*\[Zeta]
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex30 Ī³ = c ā¢ Ļµ - 1 - d ā¢ Ļµ - 2 š›¾ š‘ superscript italic-Ļµ 1 š‘‘ superscript italic-Ļµ 2 {\displaystyle{\displaystyle\gamma=c\epsilon^{-1}-d\epsilon^{-2}}}
\gamma = c\epsilon^{-1}-d\epsilon^{-2}		 

gamma = c*(epsilon)^(- 1)- d*(epsilon)^(- 2)
 
\[Gamma] == c*\[Epsilon]^(- 1)- d*\[Epsilon]^(- 2)
 Skipped - no semantic math Skipped - no semantic math - -
32.2#Ex31 Ī“ = d ā¢ Ļµ - 2 š›æ š‘‘ superscript italic-Ļµ 2 {\displaystyle{\displaystyle\delta=d\epsilon^{-2}}}
\delta = d\epsilon^{-2}		 

delta = d*(epsilon)^(- 2)
 
\[Delta] == d*\[Epsilon]^(- 2)
 Skipped - no semantic math Skipped - no semantic math - -
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