Heun Functions - 31.8 Solutions via Quadratures

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.8#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \beta-\alpha = m_{0}+\tfrac{1}{2}}
\beta-\alpha = m_{0}+\tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
beta - alpha = m[0]+(1)/(2)
 
\[Beta]- \[Alpha] == Subscript[m, 0]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma = -m_{1}+\tfrac{1}{2}}
\gamma = -m_{1}+\tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma = - m[1]+(1)/(2)
 
\[Gamma] == - Subscript[m, 1]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \delta = -m_{2}+\tfrac{1}{2}}
\delta = -m_{2}+\tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
delta = - m[2]+(1)/(2)
 
\[Delta] == - Subscript[m, 2]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \epsilon = -m_{3}+\tfrac{1}{2}}
\epsilon = -m_{3}+\tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
epsilon = - m[3]+(1)/(2)
 
\[Epsilon] == - Subscript[m, 3]+Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
31.8.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 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_{+}(\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)}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 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_{-}(\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)}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Psi_{1,2} = z^{2}+\lambda z+a}
\Psi_{1,2} = z^{2}+\lambda z+a		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \nu^{2} = (\lambda+a+1)(\lambda^{2}-4a)}
\nu^{2} = (\lambda+a+1)(\lambda^{2}-4a)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{m} = (1} 
(nu)^(2) = (lambda + a + 1)*((lambda)^(2)- 4*a)
 
\[Nu]^(2) == (\[Lambda]+ a + 1)*(\[Lambda]^(2)- 4*a)
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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)
 Skipped - no semantic math Skipped - no semantic math - -
31.8#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)
 Skipped - no semantic math Skipped - no semantic math - -
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