Heun Functions - 31.8 Solutions via Quadratures

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.8#Ex1 Ξ² - Ξ± = m 0 + 1 2 𝛽 𝛼 subscript π‘š 0 1 2 {\displaystyle{\displaystyle\beta-\alpha=m_{0}+\tfrac{1}{2}}}
\beta-\alpha = m_{0}+\tfrac{1}{2}		 

beta - alpha = m[0]+(1)/(2)
 
\[Beta]- \[Alpha] == Subscript[m, 0]+Divide[1,2]
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31.8#Ex2 Ξ³ = - m 1 + 1 2 𝛾 subscript π‘š 1 1 2 {\displaystyle{\displaystyle\gamma=-m_{1}+\tfrac{1}{2}}}
\gamma = -m_{1}+\tfrac{1}{2}		 

gamma = - m[1]+(1)/(2)
 
\[Gamma] == - Subscript[m, 1]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex3 Ξ΄ = - m 2 + 1 2 𝛿 subscript π‘š 2 1 2 {\displaystyle{\displaystyle\delta=-m_{2}+\tfrac{1}{2}}}
\delta = -m_{2}+\tfrac{1}{2}		 

delta = - m[2]+(1)/(2)
 
\[Delta] == - Subscript[m, 2]+Divide[1,2]
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31.8#Ex4 Ο΅ = - m 3 + 1 2 italic-Ο΅ subscript π‘š 3 1 2 {\displaystyle{\displaystyle\epsilon=-m_{3}+\tfrac{1}{2}}}
\epsilon = -m_{3}+\tfrac{1}{2}		 

epsilon = - m[3]+(1)/(2)
 
\[Epsilon] == - Subscript[m, 3]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8.E2 w + ⁒ ( 𝐦 ; Ξ» ; z ) = Ξ¨ g , N ⁒ ( Ξ» , z ) ⁒ exp ⁑ ( + i ⁒ Ξ½ ⁒ ( Ξ» ) 2 ⁒ ∫ z 0 z t m 1 ⁒ ( t - 1 ) m 2 ⁒ ( t - a ) m 3 ⁒ d t Ξ¨ g , N ⁒ ( Ξ» , t ) ⁒ t ⁒ ( t - 1 ) ⁒ ( t - a ) ) subscript 𝑀 𝐦 πœ† 𝑧 subscript Ξ¨ 𝑔 𝑁 πœ† 𝑧 𝑖 𝜈 πœ† 2 superscript subscript subscript 𝑧 0 𝑧 superscript 𝑑 subscript π‘š 1 superscript 𝑑 1 subscript π‘š 2 superscript 𝑑 π‘Ž subscript π‘š 3 𝑑 subscript Ξ¨ 𝑔 𝑁 πœ† 𝑑 𝑑 𝑑 1 𝑑 π‘Ž {\displaystyle{\displaystyle w_{+}(\mathbf{m};\lambda;z)=\sqrt{\Psi_{g,N}(% \lambda,z)}\*\exp\left(+\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}% (t-1)^{m_{2}}(t-a)^{m_{3}}\mathrm{d}t}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}% }\right)}}
w_{+}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{+\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}		 

w[+](m ; lambda ; z) = sqrt(Psi[g , N](lambda , z))* exp(+(I*nu(lambda))/(2)*int(((t)^(m[1])*(t - 1)^(m[2])*(t - a)^(m[3]))/(Psi[g , N](lambda , t)*sqrt(t*(t - 1)*(t - a))), t = z[0]..z))

Subscript[w, +][m ; \[Lambda]; z] == Sqrt[Subscript[\[CapitalPsi], g , N][\[Lambda], z]]* Exp[+Divide[I*\[Nu][\[Lambda]],2]*Integrate[Divide[(t)^(Subscript[m, 1])*(t - 1)^(Subscript[m, 2])*(t - a)^(Subscript[m, 3]),Subscript[\[CapitalPsi], g , N][\[Lambda], t]*Sqrt[t*(t - 1)*(t - a)]], {t, Subscript[z, 0], z}, GenerateConditions->None]]
Translation Error Translation Error - -
31.8.E2 w - ⁒ ( 𝐦 ; Ξ» ; z ) = Ξ¨ g , N ⁒ ( Ξ» , z ) ⁒ exp ⁑ ( - i ⁒ Ξ½ ⁒ ( Ξ» ) 2 ⁒ ∫ z 0 z t m 1 ⁒ ( t - 1 ) m 2 ⁒ ( t - a ) m 3 ⁒ d t Ξ¨ g , N ⁒ ( Ξ» , t ) ⁒ t ⁒ ( t - 1 ) ⁒ ( t - a ) ) subscript 𝑀 𝐦 πœ† 𝑧 subscript Ξ¨ 𝑔 𝑁 πœ† 𝑧 𝑖 𝜈 πœ† 2 superscript subscript subscript 𝑧 0 𝑧 superscript 𝑑 subscript π‘š 1 superscript 𝑑 1 subscript π‘š 2 superscript 𝑑 π‘Ž subscript π‘š 3 𝑑 subscript Ξ¨ 𝑔 𝑁 πœ† 𝑑 𝑑 𝑑 1 𝑑 π‘Ž {\displaystyle{\displaystyle w_{-}(\mathbf{m};\lambda;z)=\sqrt{\Psi_{g,N}(% \lambda,z)}\*\exp\left(-\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}% (t-1)^{m_{2}}(t-a)^{m_{3}}\mathrm{d}t}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}% }\right)}}
w_{-}(\mathbf{m};\lambda;z) = \sqrt{\Psi_{g,N}(\lambda,z)}\*\exp@{-\frac{i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\diff{t}}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}}		 

w[-](m ; lambda ; z) = sqrt(Psi[g , N](lambda , z))* exp(-(I*nu(lambda))/(2)*int(((t)^(m[1])*(t - 1)^(m[2])*(t - a)^(m[3]))/(Psi[g , N](lambda , t)*sqrt(t*(t - 1)*(t - a))), t = z[0]..z))

Subscript[w, -][m ; \[Lambda]; z] == Sqrt[Subscript[\[CapitalPsi], g , N][\[Lambda], z]]* Exp[-Divide[I*\[Nu][\[Lambda]],2]*Integrate[Divide[(t)^(Subscript[m, 1])*(t - 1)^(Subscript[m, 2])*(t - a)^(Subscript[m, 3]),Subscript[\[CapitalPsi], g , N][\[Lambda], t]*Sqrt[t*(t - 1)*(t - a)]], {t, Subscript[z, 0], z}, GenerateConditions->None]]
Translation Error Translation Error - -
31.8#Ex5 Ξ¨ 1 , 2 = z 2 + Ξ» ⁒ z + a subscript Ξ¨ 1 2 superscript 𝑧 2 πœ† 𝑧 π‘Ž {\displaystyle{\displaystyle\Psi_{1,2}=z^{2}+\lambda z+a}}
\Psi_{1,2} = z^{2}+\lambda z+a		 

Psi[1 , 2] = (z)^(2)+ lambda*z + a
 
Subscript[\[CapitalPsi], 1 , 2] == (z)^(2)+ \[Lambda]*z + a
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex6 Ξ½ 2 = ( Ξ» + a + 1 ) ⁒ ( Ξ» 2 - 4 ⁒ a ) superscript 𝜈 2 πœ† π‘Ž 1 superscript πœ† 2 4 π‘Ž {\displaystyle{\displaystyle\nu^{2}=(\lambda+a+1)(\lambda^{2}-4a)}}
\nu^{2} = (\lambda+a+1)(\lambda^{2}-4a)		 𝐦 = ( 1 fragments m fragments ( 1 {\displaystyle{\displaystyle\mathbf{m}=(1}} 
(nu)^(2) = (lambda + a + 1)*((lambda)^(2)- 4*a)
 
\[Nu]^(2) == (\[Lambda]+ a + 1)*(\[Lambda]^(2)- 4*a)
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31.8#Ex7 Ξ¨ 1 , - 1 = ( z 2 + ( Ξ» + 3 ⁒ a + 3 ) ⁒ z + a ) / z 3 subscript Ξ¨ 1 1 superscript 𝑧 2 πœ† 3 π‘Ž 3 𝑧 π‘Ž superscript 𝑧 3 {\displaystyle{\displaystyle\Psi_{1,-1}=\left(z^{2}+(\lambda+3a+3)z+a\right)/z% ^{3}}}
\Psi_{1,-1} = \left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3}		 

Psi[1 , - 1] = ((z)^(2)+(lambda + 3*a + 3)*z + a)/(z)^(3)
 
Subscript[\[CapitalPsi], 1 , - 1] == ((z)^(2)+(\[Lambda]+ 3*a + 3)*z + a)/(z)^(3)
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31.8#Ex8 Ξ½ 2 = ( Ξ» + 4 ⁒ a + 4 ) ⁒ ( ( Ξ» + 3 ⁒ a + 3 ) 2 - 4 ⁒ a ) superscript 𝜈 2 πœ† 4 π‘Ž 4 superscript πœ† 3 π‘Ž 3 2 4 π‘Ž {\displaystyle{\displaystyle\nu^{2}=(\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a% \right)}}
\nu^{2} = (\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a\right)		 𝐦 = ( 1 fragments m fragments ( 1 {\displaystyle{\displaystyle\mathbf{m}=(1}} 
(nu)^(2) = (lambda + 4*a + 4)*((lambda + 3*a + 3)^(2)- 4*a)
 
\[Nu]^(2) == (\[Lambda]+ 4*a + 4)*((\[Lambda]+ 3*a + 3)^(2)- 4*a)
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