Painlevé Transcendents - 32.7 Bäcklund Transformations

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.7.E3 W ⁢ ( ζ ; 1 2 ⁢ ε ) = 2 - 1 / 3 ⁢ ε w ⁢ ( z ; 0 ) ⁢ d d z ⁡ w ⁢ ( z ; 0 ) 𝑊 𝜁 1 2 𝜀 superscript 2 1 3 𝜀 𝑤 𝑧 0 derivative 𝑧 𝑤 𝑧 0 {\displaystyle{\displaystyle W(\zeta;\tfrac{1}{2}\varepsilon)=\frac{2^{-1/3}% \varepsilon}{w(z;0)}\frac{\mathrm{d}}{\mathrm{d}z}w(z;0)}}
W(\zeta;\tfrac{1}{2}\varepsilon) = \frac{2^{-1/3}\varepsilon}{w(z;0)}\deriv{}{z}w(z;0)		 

W(zeta ;(1)/(2)*varepsilon) = ((2)^(- 1/3)* varepsilon)/(w(z ; 0))*diff(w(z ; 0), z)

W[\[Zeta];Divide[1,2]*\[CurlyEpsilon]] == Divide[(2)^(- 1/3)* \[CurlyEpsilon],w[z ; 0]]*D[w[z ; 0], z]
Translation Error Translation Error - -
32.7.E4 w 2 ⁢ ( z ; 0 ) = 2 - 1 / 3 ⁢ ( W 2 ⁢ ( ζ ; 1 2 ⁢ ε ) - ε ⁢ d d ζ ⁡ W ⁢ ( ζ ; 1 2 ⁢ ε ) + 1 2 ⁢ ζ ) superscript 𝑤 2 𝑧 0 superscript 2 1 3 superscript 𝑊 2 𝜁 1 2 𝜀 𝜀 derivative 𝜁 𝑊 𝜁 1 2 𝜀 1 2 𝜁 {\displaystyle{\displaystyle w^{2}(z;0)=2^{-1/3}\left(W^{2}(\zeta;\tfrac{1}{2}% \varepsilon)-\varepsilon\frac{\mathrm{d}}{\mathrm{d}\zeta}W(\zeta;\tfrac{1}{2}% \varepsilon)+\tfrac{1}{2}\zeta\right)}}
w^{2}(z;0) = 2^{-1/3}\left(W^{2}(\zeta;\tfrac{1}{2}\varepsilon)-\varepsilon\deriv{}{\zeta}W(\zeta;\tfrac{1}{2}\varepsilon)+\tfrac{1}{2}\zeta\right)		 

(w(z ; 0))^(2) = (2)^(- 1/3)*((W(zeta ;(1)/(2)*varepsilon))^(2)- varepsilon*diff(W(zeta ;(1)/(2)*varepsilon)+(1)/(2)*zeta, zeta))

(w[z ; 0])^(2) == (2)^(- 1/3)*((W[\[Zeta];Divide[1,2]*\[CurlyEpsilon]])^(2)- \[CurlyEpsilon]*D[W[\[Zeta];Divide[1,2]*\[CurlyEpsilon]]+Divide[1,2]*\[Zeta], \[Zeta]])
Translation Error Translation Error - -
32.7.E5 α + 1 2 w α + 1 + w α + α - 1 2 w α + w α - 1 + 2 ⁢ w α 2 + z = 0 𝛼 1 2 subscript 𝑤 𝛼 1 subscript 𝑤 𝛼 𝛼 1 2 subscript 𝑤 𝛼 subscript 𝑤 𝛼 1 2 superscript subscript 𝑤 𝛼 2 𝑧 0 {\displaystyle{\displaystyle\frac{\alpha+\tfrac{1}{2}}{w_{\alpha+1}+w_{\alpha}% }+\frac{\alpha-\tfrac{1}{2}}{w_{\alpha}+w_{\alpha-1}}+2w_{\alpha}^{2}+z=0}}
\frac{\alpha+\tfrac{1}{2}}{w_{\alpha+1}+w_{\alpha}}+\frac{\alpha-\tfrac{1}{2}}{w_{\alpha}+w_{\alpha-1}}+2w_{\alpha}^{2}+z = 0		 

(alpha +(1)/(2))/(w[alpha + 1]+ w[alpha])+(alpha -(1)/(2))/(w[alpha]+ w[alpha - 1])+ 2*(w[alpha])^(2)+ z = 0
 
Divide[\[Alpha]+Divide[1,2],Subscript[w, \[Alpha]+ 1]+ Subscript[w, \[Alpha]]]+Divide[\[Alpha]-Divide[1,2],Subscript[w, \[Alpha]]+ Subscript[w, \[Alpha]- 1]]+ 2*(Subscript[w, \[Alpha]])^(2)+ z == 0
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32.7.E6 ( α 1 , β 1 , γ 1 , δ 1 ) = ( - α 0 , - β 0 , γ 0 , δ 0 ) subscript 𝛼 1 subscript 𝛽 1 subscript 𝛾 1 subscript 𝛿 1 subscript 𝛼 0 subscript 𝛽 0 subscript 𝛾 0 subscript 𝛿 0 {\displaystyle{\displaystyle(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(-% \alpha_{0},-\beta_{0},\gamma_{0},\delta_{0})}}
(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1}) = (-\alpha_{0},-\beta_{0},\gamma_{0},\delta_{0})		 

(alpha[1], beta[1], gamma[1], delta[1]) = (- alpha[0], - beta[0], gamma[0], delta[0])
 
(Subscript[\[Alpha], 1], Subscript[\[Beta], 1], Subscript[\[Gamma], 1], Subscript[\[Delta], 1]) == (- Subscript[\[Alpha], 0], - Subscript[\[Beta], 0], Subscript[\[Gamma], 0], Subscript[\[Delta], 0])
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32.7.E7 ( α 2 , β 2 , γ 2 , δ 2 ) = ( - β 0 , - α 0 , - δ 0 , - γ 0 ) subscript 𝛼 2 subscript 𝛽 2 subscript 𝛾 2 subscript 𝛿 2 subscript 𝛽 0 subscript 𝛼 0 subscript 𝛿 0 subscript 𝛾 0 {\displaystyle{\displaystyle(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(-% \beta_{0},-\alpha_{0},-\delta_{0},-\gamma_{0})}}
(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2}) = (-\beta_{0},-\alpha_{0},-\delta_{0},-\gamma_{0})		 

(alpha[2], beta[2], gamma[2], delta[2]) = (- beta[0], - alpha[0], - delta[0], - gamma[0])
 
(Subscript[\[Alpha], 2], Subscript[\[Beta], 2], Subscript[\[Gamma], 2], Subscript[\[Delta], 2]) == (- Subscript[\[Beta], 0], - Subscript[\[Alpha], 0], - Subscript[\[Delta], 0], - Subscript[\[Gamma], 0])
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32.7#Ex1 α 1 = α 3 subscript 𝛼 1 subscript 𝛼 3 {\displaystyle{\displaystyle\alpha_{1}=\alpha_{3}}}
\alpha_{1} = \alpha_{3}		 

alpha[1] = alpha[3]
 
Subscript[\[Alpha], 1] == Subscript[\[Alpha], 3]
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32.7#Ex2 α 2 = α 4 subscript 𝛼 2 subscript 𝛼 4 {\displaystyle{\displaystyle\alpha_{2}=\alpha_{4}}}
\alpha_{2} = \alpha_{4}		 

alpha[2] = alpha[4]
 
Subscript[\[Alpha], 2] == Subscript[\[Alpha], 4]
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32.7#Ex3 β 1 = β 2 subscript 𝛽 1 subscript 𝛽 2 {\displaystyle{\displaystyle\beta_{1}=\beta_{2}}}
\beta_{1} = \beta_{2}		 

beta[1] = beta[2]
 
Subscript[\[Beta], 1] == Subscript[\[Beta], 2]
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32.7#Ex4 β 3 = β 4 subscript 𝛽 3 subscript 𝛽 4 {\displaystyle{\displaystyle\beta_{3}=\beta_{4}}}
\beta_{3} = \beta_{4}		 

beta[3] = beta[4]
 
Subscript[\[Beta], 3] == Subscript[\[Beta], 4]
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32.7#Ex5 β 5 = β 0 + 2 subscript 𝛽 5 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{5}=\beta_{0}+2}}
\beta_{5} = \beta_{0}+2		 

beta[5] = beta[0]+ 2
 
Subscript[\[Beta], 5] == Subscript[\[Beta], 0]+ 2
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32.7#Ex6 β 6 = β 0 - 2 subscript 𝛽 6 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{6}=\beta_{0}-2}}
\beta_{6} = \beta_{0}-2		 

beta[6] = beta[0]- 2
 
Subscript[\[Beta], 6] == Subscript[\[Beta], 0]- 2
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32.7#Ex7 w ⁢ ( z ; a , b , 0 , 0 ) = W 2 ⁢ ( ζ ; 0 , 0 , a , b ) 𝑤 𝑧 𝑎 𝑏 0 0 superscript 𝑊 2 𝜁 0 0 𝑎 𝑏 {\displaystyle{\displaystyle w(z;a,b,0,0)=W^{2}(\zeta;0,0,a,b)}}
w(z;a,b,0,0) = W^{2}(\zeta;0,0,a,b)		 

w(z ; a , b , 0 , 0) = (W(zeta ; 0 , 0 , a , b))^(2)
 
w[z ; a , b , 0 , 0] == (W[\[Zeta]; 0 , 0 , a , b])^(2)
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32.7#Ex8 z = 1 2 ⁢ ζ 2 𝑧 1 2 superscript 𝜁 2 {\displaystyle{\displaystyle z=\tfrac{1}{2}\zeta^{2}}}
z = \tfrac{1}{2}\zeta^{2}		 

z = (1)/(2)*(zeta)^(2)
 
z == Divide[1,2]*\[Zeta]^(2)
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32.7#Ex9 α 1 + = 1 4 ⁢ ( 2 - 2 ⁢ α 0 + 3 ⁢ - 2 ⁢ β 0 ) superscript subscript 𝛼 1 1 4 2 2 subscript 𝛼 0 3 2 subscript 𝛽 0 {\displaystyle{\displaystyle\alpha_{1}^{+}=\tfrac{1}{4}\left(2-2\alpha_{0}+3% \sqrt{-2\beta_{0}}\right)}}
\alpha_{1}^{+} = \tfrac{1}{4}\left(2-2\alpha_{0}+ 3\sqrt{-2\beta_{0}}\right)		 

(alpha[1])^(+) = (1)/(4)*(2 - 2*alpha[0]+ 3*sqrt(- 2*beta[0]))
 
(Subscript[\[Alpha], 1])^(+) == Divide[1,4]*(2 - 2*Subscript[\[Alpha], 0]+ 3*Sqrt[- 2*Subscript[\[Beta], 0]])
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32.7#Ex10 β 1 + = - 1 2 ⁢ ( 1 + α 0 + 1 2 ⁢ - 2 ⁢ β 0 ) 2 superscript subscript 𝛽 1 1 2 superscript 1 subscript 𝛼 0 1 2 2 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{1}^{+}=-\tfrac{1}{2}\left(1+\alpha_{0}+% \tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}}}
\beta_{1}^{+} = -\tfrac{1}{2}\left(1+\alpha_{0}+\tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}		 

(beta[1])^(+) = -(1)/(2)*(1 + alpha[0]+(1)/(2)*sqrt(- 2*beta[0]))^(2)
 
(Subscript[\[Beta], 1])^(+) == -Divide[1,2]*(1 + Subscript[\[Alpha], 0]+Divide[1,2]*Sqrt[- 2*Subscript[\[Beta], 0]])^(2)
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32.7#Ex11 α 2 + = - 1 4 ⁢ ( 2 + 2 ⁢ α 0 + 3 ⁢ - 2 ⁢ β 0 ) superscript subscript 𝛼 2 1 4 2 2 subscript 𝛼 0 3 2 subscript 𝛽 0 {\displaystyle{\displaystyle\alpha_{2}^{+}=-\tfrac{1}{4}\left(2+2\alpha_{0}+3% \sqrt{-2\beta_{0}}\right)}}
\alpha_{2}^{+} = -\tfrac{1}{4}\left(2+2\alpha_{0}+ 3\sqrt{-2\beta_{0}}\right)		 

(alpha[2])^(+) = -(1)/(4)*(2 + 2*alpha[0]+ 3*sqrt(- 2*beta[0]))
 
(Subscript[\[Alpha], 2])^(+) == -Divide[1,4]*(2 + 2*Subscript[\[Alpha], 0]+ 3*Sqrt[- 2*Subscript[\[Beta], 0]])
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32.7#Ex12 β 2 + = - 1 2 ⁢ ( 1 - α 0 + 1 2 ⁢ - 2 ⁢ β 0 ) 2 superscript subscript 𝛽 2 1 2 superscript 1 subscript 𝛼 0 1 2 2 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{2}^{+}=-\tfrac{1}{2}\left(1-\alpha_{0}+% \tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}}}
\beta_{2}^{+} = -\tfrac{1}{2}\left(1-\alpha_{0}+\tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}		 

(beta[2])^(+) = -(1)/(2)*(1 - alpha[0]+(1)/(2)*sqrt(- 2*beta[0]))^(2)
 
(Subscript[\[Beta], 2])^(+) == -Divide[1,2]*(1 - Subscript[\[Alpha], 0]+Divide[1,2]*Sqrt[- 2*Subscript[\[Beta], 0]])^(2)
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32.7#Ex13 α 3 + = 3 2 - 1 2 ⁢ α 0 - 3 4 ⁢ - 2 ⁢ β 0 superscript subscript 𝛼 3 3 2 1 2 subscript 𝛼 0 3 4 2 subscript 𝛽 0 {\displaystyle{\displaystyle\alpha_{3}^{+}=\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}% -\tfrac{3}{4}\sqrt{-2\beta_{0}}}}
\alpha_{3}^{+} = \tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}-\tfrac{3}{4}\sqrt{-2\beta_{0}}		 

(alpha[3])^(+) = (3)/(2)-(1)/(2)*alpha[0]-(3)/(4)*sqrt(- 2*beta[0])
 
(Subscript[\[Alpha], 3])^(+) == Divide[3,2]-Divide[1,2]*Subscript[\[Alpha], 0]-Divide[3,4]*Sqrt[- 2*Subscript[\[Beta], 0]]
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32.7#Ex14 β 3 + = - 1 2 ⁢ ( 1 - α 0 + 1 2 ⁢ - 2 ⁢ β 0 ) 2 superscript subscript 𝛽 3 1 2 superscript 1 subscript 𝛼 0 1 2 2 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{3}^{+}=-\tfrac{1}{2}\left(1-\alpha_{0}+% \tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}}}
\beta_{3}^{+} = -\tfrac{1}{2}\left(1-\alpha_{0}+\tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}		 

(beta[3])^(+) = -(1)/(2)*(1 - alpha[0]+(1)/(2)*sqrt(- 2*beta[0]))^(2)
 
(Subscript[\[Beta], 3])^(+) == -Divide[1,2]*(1 - Subscript[\[Alpha], 0]+Divide[1,2]*Sqrt[- 2*Subscript[\[Beta], 0]])^(2)
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32.7#Ex15 α 4 + = - 3 2 - 1 2 ⁢ α 0 - 3 4 ⁢ - 2 ⁢ β 0 superscript subscript 𝛼 4 3 2 1 2 subscript 𝛼 0 3 4 2 subscript 𝛽 0 {\displaystyle{\displaystyle\alpha_{4}^{+}=-\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0% }-\tfrac{3}{4}\sqrt{-2\beta_{0}}}}
\alpha_{4}^{+} = -\tfrac{3}{2}-\tfrac{1}{2}\alpha_{0}-\tfrac{3}{4}\sqrt{-2\beta_{0}}		 

(alpha[4])^(+) = -(3)/(2)-(1)/(2)*alpha[0]-(3)/(4)*sqrt(- 2*beta[0])
 
(Subscript[\[Alpha], 4])^(+) == -Divide[3,2]-Divide[1,2]*Subscript[\[Alpha], 0]-Divide[3,4]*Sqrt[- 2*Subscript[\[Beta], 0]]
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32.7#Ex16 β 4 + = - 1 2 ⁢ ( - 1 - α 0 + 1 2 ⁢ - 2 ⁢ β 0 ) 2 superscript subscript 𝛽 4 1 2 superscript 1 subscript 𝛼 0 1 2 2 subscript 𝛽 0 2 {\displaystyle{\displaystyle\beta_{4}^{+}=-\tfrac{1}{2}\left(-1-\alpha_{0}+% \tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}}}
\beta_{4}^{+} = -\tfrac{1}{2}\left(-1-\alpha_{0}+\tfrac{1}{2}\sqrt{-2\beta_{0}}\right)^{2}		 

(beta[4])^(+) = -(1)/(2)*(- 1 - alpha[0]+(1)/(2)*sqrt(- 2*beta[0]))^(2)
 
(Subscript[\[Beta], 4])^(+) == -Divide[1,2]*(- 1 - Subscript[\[Alpha], 0]+Divide[1,2]*Sqrt[- 2*Subscript[\[Beta], 0]])^(2)
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32.7#Ex17 z 1 = - z 0 subscript 𝑧 1 subscript 𝑧 0 {\displaystyle{\displaystyle z_{1}=-z_{0}}}
z_{1} = -z_{0}		 

z[1] = - z[0]
 
Subscript[z, 1] == - Subscript[z, 0]
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32.7#Ex18 z 2 = z 0 subscript 𝑧 2 subscript 𝑧 0 {\displaystyle{\displaystyle z_{2}=z_{0}}}
z_{2} = z_{0}		 

z[2] = z[0]
 
Subscript[z, 2] == Subscript[z, 0]
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32.7#Ex19 ( α 1 , β 1 , γ 1 , δ 1 ) = ( α 0 , β 0 , - γ 0 , δ 0 ) subscript 𝛼 1 subscript 𝛽 1 subscript 𝛾 1 subscript 𝛿 1 subscript 𝛼 0 subscript 𝛽 0 subscript 𝛾 0 subscript 𝛿 0 {\displaystyle{\displaystyle(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(% \alpha_{0},\beta_{0},-\gamma_{0},\delta_{0})}}
(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1}) = (\alpha_{0},\beta_{0},-\gamma_{0},\delta_{0})		 

(alpha[1], beta[1], gamma[1], delta[1]) = (alpha[0], beta[0], - gamma[0], delta[0])
 
(Subscript[\[Alpha], 1], Subscript[\[Beta], 1], Subscript[\[Gamma], 1], Subscript[\[Delta], 1]) == (Subscript[\[Alpha], 0], Subscript[\[Beta], 0], - Subscript[\[Gamma], 0], Subscript[\[Delta], 0])
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32.7#Ex20 ( α 2 , β 2 , γ 2 , δ 2 ) = ( - β 0 , - α 0 , - γ 0 , δ 0 ) subscript 𝛼 2 subscript 𝛽 2 subscript 𝛾 2 subscript 𝛿 2 subscript 𝛽 0 subscript 𝛼 0 subscript 𝛾 0 subscript 𝛿 0 {\displaystyle{\displaystyle(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(-% \beta_{0},-\alpha_{0},-\gamma_{0},\delta_{0})}}
(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2}) = (-\beta_{0},-\alpha_{0},-\gamma_{0},\delta_{0})		 

(alpha[2], beta[2], gamma[2], delta[2]) = (- beta[0], - alpha[0], - gamma[0], delta[0])
 
(Subscript[\[Alpha], 2], Subscript[\[Beta], 2], Subscript[\[Gamma], 2], Subscript[\[Delta], 2]) == (- Subscript[\[Beta], 0], - Subscript[\[Alpha], 0], - Subscript[\[Gamma], 0], Subscript[\[Delta], 0])
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32.7#Ex21 α 1 = 1 8 ⁢ ( γ 0 + ε 1 ⁢ ( 1 - ε 3 ⁢ - 2 ⁢ β 0 - ε 2 ⁢ 2 ⁢ α 0 ) ) 2 subscript 𝛼 1 1 8 superscript subscript 𝛾 0 subscript 𝜀 1 1 subscript 𝜀 3 2 subscript 𝛽 0 subscript 𝜀 2 2 subscript 𝛼 0 2 {\displaystyle{\displaystyle\alpha_{1}=\tfrac{1}{8}\left(\gamma_{0}+% \varepsilon_{1}\left(1-\varepsilon_{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{% 2\alpha_{0}}\right)\right)^{2}}}
\alpha_{1} = \tfrac{1}{8}\left(\gamma_{0}+\varepsilon_{1}\left(1-\varepsilon_{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2}		 

alpha[1] = (1)/(8)*(gamma[0]+ varepsilon[1]*(1 - varepsilon[3]*sqrt(- 2*beta[0])- varepsilon[2]*sqrt(2*alpha[0])))^(2)
 
Subscript[\[Alpha], 1] == Divide[1,8]*(Subscript[\[Gamma], 0]+ Subscript[\[CurlyEpsilon], 1]*(1 - Subscript[\[CurlyEpsilon], 3]*Sqrt[- 2*Subscript[\[Beta], 0]]- Subscript[\[CurlyEpsilon], 2]*Sqrt[2*Subscript[\[Alpha], 0]]))^(2)
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32.7#Ex22 β 1 = - 1 8 ⁢ ( γ 0 - ε 1 ⁢ ( 1 - ε 3 ⁢ - 2 ⁢ β 0 - ε 2 ⁢ 2 ⁢ α 0 ) ) 2 subscript 𝛽 1 1 8 superscript subscript 𝛾 0 subscript 𝜀 1 1 subscript 𝜀 3 2 subscript 𝛽 0 subscript 𝜀 2 2 subscript 𝛼 0 2 {\displaystyle{\displaystyle\beta_{1}=-\tfrac{1}{8}\left(\gamma_{0}-% \varepsilon_{1}\left(1-\varepsilon_{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{% 2\alpha_{0}}\right)\right)^{2}}}
\beta_{1} = -\tfrac{1}{8}\left(\gamma_{0}-\varepsilon_{1}\left(1-\varepsilon_{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)\right)^{2}		 

beta[1] = -(1)/(8)*(gamma[0]- varepsilon[1]*(1 - varepsilon[3]*sqrt(- 2*beta[0])- varepsilon[2]*sqrt(2*alpha[0])))^(2)
 
Subscript[\[Beta], 1] == -Divide[1,8]*(Subscript[\[Gamma], 0]- Subscript[\[CurlyEpsilon], 1]*(1 - Subscript[\[CurlyEpsilon], 3]*Sqrt[- 2*Subscript[\[Beta], 0]]- Subscript[\[CurlyEpsilon], 2]*Sqrt[2*Subscript[\[Alpha], 0]]))^(2)
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32.7#Ex23 γ 1 = ε 1 ⁢ ( ε 3 ⁢ - 2 ⁢ β 0 - ε 2 ⁢ 2 ⁢ α 0 ) subscript 𝛾 1 subscript 𝜀 1 subscript 𝜀 3 2 subscript 𝛽 0 subscript 𝜀 2 2 subscript 𝛼 0 {\displaystyle{\displaystyle\gamma_{1}=\varepsilon_{1}\left(\varepsilon_{3}% \sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)}}
\gamma_{1} = \varepsilon_{1}\left(\varepsilon_{3}\sqrt{-2\beta_{0}}-\varepsilon_{2}\sqrt{2\alpha_{0}}\right)		 

gamma[1] = varepsilon[1]*(varepsilon[3]*sqrt(- 2*beta[0])- varepsilon[2]*sqrt(2*alpha[0]))
 
Subscript[\[Gamma], 1] == Subscript[\[CurlyEpsilon], 1]*(Subscript[\[CurlyEpsilon], 3]*Sqrt[- 2*Subscript[\[Beta], 0]]- Subscript[\[CurlyEpsilon], 2]*Sqrt[2*Subscript[\[Alpha], 0]])
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32.7#Ex24 W ⁢ ( ζ ; α 0 , β 0 , γ 0 , δ 0 ) = v - 1 v + 1 𝑊 𝜁 subscript 𝛼 0 subscript 𝛽 0 subscript 𝛾 0 subscript 𝛿 0 𝑣 1 𝑣 1 {\displaystyle{\displaystyle W(\zeta;\alpha_{0},\beta_{0},\gamma_{0},\delta_{0% })=\frac{v-1}{v+1}}}
W(\zeta;\alpha_{0},\beta_{0},\gamma_{0},\delta_{0}) = \frac{v-1}{v+1}		 

W(zeta ; alpha[0], beta[0], gamma[0], delta[0]) = (v - 1)/(v + 1)
 
W[\[Zeta]; Subscript[\[Alpha], 0], Subscript[\[Beta], 0], Subscript[\[Gamma], 0], Subscript[\[Delta], 0]] == Divide[v - 1,v + 1]
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32.7#Ex25 z = 2 ⁢ ζ 𝑧 2 𝜁 {\displaystyle{\displaystyle z=\sqrt{2\zeta}}}
z = \sqrt{2\zeta}		 

z = sqrt(2*zeta)
 
z == Sqrt[2*\[Zeta]]
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32.7.E32 ( α 0 , β 0 , γ 0 , δ 0 ) = ( ( β - ε ⁢ α + 2 ) 2 / 32 , - ( β + ε ⁢ α - 2 ) 2 / 32 , - ε , 0 ) subscript 𝛼 0 subscript 𝛽 0 subscript 𝛾 0 subscript 𝛿 0 superscript 𝛽 𝜀 𝛼 2 2 32 superscript 𝛽 𝜀 𝛼 2 2 32 𝜀 0 {\displaystyle{\displaystyle(\alpha_{0},\beta_{0},\gamma_{0},\delta_{0})={% \left((\beta-\varepsilon\alpha+2)^{2}/32,-(\beta+\varepsilon\alpha-2)^{2}/32,-% \varepsilon,0\right)}}}
(\alpha_{0},\beta_{0},\gamma_{0},\delta_{0}) = {\left((\beta-\varepsilon\alpha+2)^{2}/32,-(\beta+\varepsilon\alpha-2)^{2}/32,-\varepsilon,0\right)}		 

(alpha[0], beta[0], gamma[0], delta[0]) = ((beta - varepsilon*alpha + 2)^(2)/32 , -(beta + varepsilon*alpha - 2)^(2)/32 , - varepsilon , 0)
 
(Subscript[\[Alpha], 0], Subscript[\[Beta], 0], Subscript[\[Gamma], 0], Subscript[\[Delta], 0]) == ((\[Beta]- \[CurlyEpsilon]*\[Alpha]+ 2)^(2)/32 , -(\[Beta]+ \[CurlyEpsilon]*\[Alpha]- 2)^(2)/32 , - \[CurlyEpsilon], 0)
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32.7.E33 z 1 = 1 / z 0 subscript 𝑧 1 1 subscript 𝑧 0 {\displaystyle{\displaystyle z_{1}=1/z_{0}}}
z_{1} = 1/z_{0}		 

z[1] = 1/z[0]
 
Subscript[z, 1] == 1/Subscript[z, 0]
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32.7.E34 z 2 = 1 - z 0 subscript 𝑧 2 1 subscript 𝑧 0 {\displaystyle{\displaystyle z_{2}=1-z_{0}}}
z_{2} = 1-z_{0}		 

z[2] = 1 - z[0]
 
Subscript[z, 2] == 1 - Subscript[z, 0]
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32.7.E35 z 3 = 1 / z 0 subscript 𝑧 3 1 subscript 𝑧 0 {\displaystyle{\displaystyle z_{3}=1/z_{0}}}
z_{3} = 1/z_{0}		 

z[3] = 1/z[0]
 
Subscript[z, 3] == 1/Subscript[z, 0]
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32.7.E36 ( α 1 , β 1 , γ 1 , δ 1 ) = ( α 0 , β 0 , - δ 0 + 1 2 , - γ 0 + 1 2 ) subscript 𝛼 1 subscript 𝛽 1 subscript 𝛾 1 subscript 𝛿 1 subscript 𝛼 0 subscript 𝛽 0 subscript 𝛿 0 1 2 subscript 𝛾 0 1 2 {\displaystyle{\displaystyle(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})=(% \alpha_{0},\beta_{0},-\delta_{0}+\tfrac{1}{2},-\gamma_{0}+\tfrac{1}{2})}}
(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1}) = (\alpha_{0},\beta_{0},-\delta_{0}+\tfrac{1}{2},-\gamma_{0}+\tfrac{1}{2})		 

(alpha[1], beta[1], gamma[1], delta[1]) = (alpha[0], beta[0], - delta[0]+(1)/(2), - gamma[0]+(1)/(2))
 
(Subscript[\[Alpha], 1], Subscript[\[Beta], 1], Subscript[\[Gamma], 1], Subscript[\[Delta], 1]) == (Subscript[\[Alpha], 0], Subscript[\[Beta], 0], - Subscript[\[Delta], 0]+Divide[1,2], - Subscript[\[Gamma], 0]+Divide[1,2])
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32.7.E37 ( α 2 , β 2 , γ 2 , δ 2 ) = ( α 0 , - γ 0 , - β 0 , δ 0 ) subscript 𝛼 2 subscript 𝛽 2 subscript 𝛾 2 subscript 𝛿 2 subscript 𝛼 0 subscript 𝛾 0 subscript 𝛽 0 subscript 𝛿 0 {\displaystyle{\displaystyle(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2})=(% \alpha_{0},-\gamma_{0},-\beta_{0},\delta_{0})}}
(\alpha_{2},\beta_{2},\gamma_{2},\delta_{2}) = (\alpha_{0},-\gamma_{0},-\beta_{0},\delta_{0})		 

(alpha[2], beta[2], gamma[2], delta[2]) = (alpha[0], - gamma[0], - beta[0], delta[0])
 
(Subscript[\[Alpha], 2], Subscript[\[Beta], 2], Subscript[\[Gamma], 2], Subscript[\[Delta], 2]) == (Subscript[\[Alpha], 0], - Subscript[\[Gamma], 0], - Subscript[\[Beta], 0], Subscript[\[Delta], 0])
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32.7.E38 ( α 3 , β 3 , γ 3 , δ 3 ) = ( - β 0 , - α 0 , γ 0 , δ 0 ) subscript 𝛼 3 subscript 𝛽 3 subscript 𝛾 3 subscript 𝛿 3 subscript 𝛽 0 subscript 𝛼 0 subscript 𝛾 0 subscript 𝛿 0 {\displaystyle{\displaystyle(\alpha_{3},\beta_{3},\gamma_{3},\delta_{3})=(-% \beta_{0},-\alpha_{0},\gamma_{0},\delta_{0})}}
(\alpha_{3},\beta_{3},\gamma_{3},\delta_{3}) = (-\beta_{0},-\alpha_{0},\gamma_{0},\delta_{0})		 

(alpha[3], beta[3], gamma[3], delta[3]) = (- beta[0], - alpha[0], gamma[0], delta[0])
 
(Subscript[\[Alpha], 3], Subscript[\[Beta], 3], Subscript[\[Gamma], 3], Subscript[\[Delta], 3]) == (- Subscript[\[Beta], 0], - Subscript[\[Alpha], 0], Subscript[\[Gamma], 0], Subscript[\[Delta], 0])
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32.7.E42 ( α , β , γ , δ ) = ( 1 2 ⁢ ( θ ∞ - 1 ) 2 , - 1 2 ⁢ θ 0 2 , 1 2 ⁢ θ 1 2 , 1 2 ⁢ ( 1 - θ 2 2 ) ) 𝛼 𝛽 𝛾 𝛿 1 2 superscript subscript 𝜃 1 2 1 2 superscript subscript 𝜃 0 2 1 2 superscript subscript 𝜃 1 2 1 2 1 superscript subscript 𝜃 2 2 {\displaystyle{\displaystyle(\alpha,\beta,\gamma,\delta)=\left(\tfrac{1}{2}(% \theta_{\infty}-1)^{2},-\tfrac{1}{2}\theta_{0}^{2},\tfrac{1}{2}\theta_{1}^{2},% \tfrac{1}{2}(1-\theta_{2}^{2})\right)}}
(\alpha,\beta,\gamma,\delta) = \left(\tfrac{1}{2}(\theta_{\infty}-1)^{2},-\tfrac{1}{2}\theta_{0}^{2},\tfrac{1}{2}\theta_{1}^{2},\tfrac{1}{2}(1-\theta_{2}^{2})\right)		 

(alpha , beta , gamma , delta) = ((1)/(2)*(theta[infinity]- 1)^(2), -(1)/(2)*(theta[0])^(2),(1)/(2)*(theta[1])^(2),(1)/(2)*(1 - (theta[2])^(2)))
 
(\[Alpha], \[Beta], \[Gamma], \[Delta]) == (Divide[1,2]*(Subscript[\[Theta], Infinity]- 1)^(2), -Divide[1,2]*(Subscript[\[Theta], 0])^(2),Divide[1,2]*(Subscript[\[Theta], 1])^(2),Divide[1,2]*(1 - (Subscript[\[Theta], 2])^(2)))
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32.7.E43 ( A , B , C , D ) = ( 1 2 ⁢ ( Θ ∞ - 1 ) 2 , - 1 2 ⁢ Θ 0 2 , 1 2 ⁢ Θ 1 2 , 1 2 ⁢ ( 1 - Θ 2 2 ) ) 𝐴 𝐵 𝐶 𝐷 1 2 superscript subscript Θ 1 2 1 2 superscript subscript Θ 0 2 1 2 superscript subscript Θ 1 2 1 2 1 superscript subscript Θ 2 2 {\displaystyle{\displaystyle(A,B,C,D)=\left(\tfrac{1}{2}(\Theta_{\infty}-1)^{2% },-\tfrac{1}{2}\Theta_{0}^{2},\tfrac{1}{2}\Theta_{1}^{2},\tfrac{1}{2}(1-\Theta% _{2}^{2})\right)}}
(A,B,C,D) = \left(\tfrac{1}{2}(\Theta_{\infty}-1)^{2},-\tfrac{1}{2}\Theta_{0}^{2},\tfrac{1}{2}\Theta_{1}^{2},\tfrac{1}{2}(1-\Theta_{2}^{2})\right)		 

(A , B , C , D) = ((1)/(2)*(Theta[infinity]- 1)^(2), -(1)/(2)*(Theta[0])^(2),(1)/(2)*(Theta[1])^(2),(1)/(2)*(1 - (Theta[2])^(2)))
 
(A , B , C , D) == (Divide[1,2]*(Subscript[\[CapitalTheta], Infinity]- 1)^(2), -Divide[1,2]*(Subscript[\[CapitalTheta], 0])^(2),Divide[1,2]*(Subscript[\[CapitalTheta], 1])^(2),Divide[1,2]*(1 - (Subscript[\[CapitalTheta], 2])^(2)))
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32.7.E44 θ j = Θ j + 1 2 ⁢ σ subscript 𝜃 𝑗 subscript Θ 𝑗 1 2 𝜎 {\displaystyle{\displaystyle\theta_{j}=\Theta_{j}+\tfrac{1}{2}\sigma}}
\theta_{j} = \Theta_{j}+\tfrac{1}{2}\sigma		 

theta[j] = Theta[j]+(1)/(2)*sigma
 
Subscript[\[Theta], j] == Subscript[\[CapitalTheta], j]+Divide[1,2]*\[Sigma]
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32.7.E45 σ = θ 0 + θ 1 + θ 2 + θ ∞ - 1 𝜎 subscript 𝜃 0 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 {\displaystyle{\displaystyle\sigma=\theta_{0}+\theta_{1}+\theta_{2}+\theta_{% \infty}-1}}
\sigma = \theta_{0}+\theta_{1}+\theta_{2}+\theta_{\infty}-1		 

sigma = theta[0]+ theta[1]+ theta[2]+ theta[infinity]- 1
 
\[Sigma] == Subscript[\[Theta], 0]+ Subscript[\[Theta], 1]+ Subscript[\[Theta], 2]+ Subscript[\[Theta], Infinity]- 1
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32.7#Ex26 u 1 ⁢ ( ζ 1 ) = ( 1 - w ) ⁢ ( w - z ) ( 1 + z ) 2 ⁢ w subscript 𝑢 1 subscript 𝜁 1 1 𝑤 𝑤 𝑧 superscript 1 𝑧 2 𝑤 {\displaystyle{\displaystyle u_{1}(\zeta_{1})=\frac{(1-w)(w-z)}{(1+\sqrt{z})^{% 2}w}}}
u_{1}(\zeta_{1}) = \frac{(1-w)(w-z)}{(1+\sqrt{z})^{2}w}		 

u[1](zeta[1]) = ((1 - w)*(w - z))/((1 +sqrt(z))^(2)* w)
 
Subscript[u, 1][Subscript[\[Zeta], 1]] == Divide[(1 - w)*(w - z),(1 +Sqrt[z])^(2)* w]
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32.7#Ex27 ζ 1 = ( 1 - z 1 + z ) 2 subscript 𝜁 1 superscript 1 𝑧 1 𝑧 2 {\displaystyle{\displaystyle\zeta_{1}=\left(\frac{1-\sqrt{z}}{1+\sqrt{z}}% \right)^{2}}}
\zeta_{1} = \left(\frac{1-\sqrt{z}}{1+\sqrt{z}}\right)^{2}		 

zeta[1] = ((1 -sqrt(z))/(1 +sqrt(z)))^(2)
 
Subscript[\[Zeta], 1] == (Divide[1 -Sqrt[z],1 +Sqrt[z]])^(2)
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32.7#Ex28 u 2 ⁢ ( ζ 2 ) = ( w 2 - z ) 2 4 ⁢ w ⁢ ( w - 1 ) ⁢ ( w - z ) subscript 𝑢 2 subscript 𝜁 2 superscript superscript 𝑤 2 𝑧 2 4 𝑤 𝑤 1 𝑤 𝑧 {\displaystyle{\displaystyle u_{2}(\zeta_{2})=\frac{(w^{2}-z)^{2}}{4w(w-1)(w-z% )}}}
u_{2}(\zeta_{2}) = \frac{(w^{2}-z)^{2}}{4w(w-1)(w-z)}		 

u[2](zeta[2]) = (((w)^(2)- z)^(2))/(4*w*(w - 1)*(w - z))
 
Subscript[u, 2][Subscript[\[Zeta], 2]] == Divide[((w)^(2)- z)^(2),4*w*(w - 1)*(w - z)]
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32.7#Ex29 ζ 2 = z subscript 𝜁 2 𝑧 {\displaystyle{\displaystyle\zeta_{2}=z}}
\zeta_{2} = z		 

zeta[2] = z
 
Subscript[\[Zeta], 2] == z
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32.7.E49 u 3 ⁢ ( ζ 3 ) = ( 1 - z 1 / 4 1 + z 1 / 4 ) 2 ⁢ ( w + z 1 / 4 w - z 1 / 4 ) 2 subscript 𝑢 3 subscript 𝜁 3 superscript 1 superscript 𝑧 1 4 1 superscript 𝑧 1 4 2 superscript 𝑤 superscript 𝑧 1 4 𝑤 superscript 𝑧 1 4 2 {\displaystyle{\displaystyle u_{3}(\zeta_{3})=\left(\frac{1-z^{1/4}}{1+z^{1/4}% }\right)^{2}\left(\frac{\sqrt{w}+z^{1/4}}{\sqrt{w}-z^{1/4}}\right)^{2}}}
u_{3}(\zeta_{3}) = \left(\frac{1-z^{1/4}}{1+z^{1/4}}\right)^{2}\left(\frac{\sqrt{w}+z^{1/4}}{\sqrt{w}-z^{1/4}}\right)^{2}		 

u[3](zeta[3]) = ((1 - (z)^(1/4))/(1 + (z)^(1/4)))^(2)*((sqrt(w)+ (z)^(1/4))/(sqrt(w)- (z)^(1/4)))^(2)
 
Subscript[u, 3][Subscript[\[Zeta], 3]] == (Divide[1 - (z)^(1/4),1 + (z)^(1/4)])^(2)*(Divide[Sqrt[w]+ (z)^(1/4),Sqrt[w]- (z)^(1/4)])^(2)
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32.7.E50 ζ 3 = ( 1 - z 1 / 4 1 + z 1 / 4 ) 4 subscript 𝜁 3 superscript 1 superscript 𝑧 1 4 1 superscript 𝑧 1 4 4 {\displaystyle{\displaystyle\zeta_{3}=\left(\frac{1-z^{1/4}}{1+z^{1/4}}\right)% ^{4}}}
\zeta_{3} = \left(\frac{1-z^{1/4}}{1+z^{1/4}}\right)^{4}		 

zeta[3] = ((1 - (z)^(1/4))/(1 + (z)^(1/4)))^(4)
 
Subscript[\[Zeta], 3] == (Divide[1 - (z)^(1/4),1 + (z)^(1/4)])^(4)
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