Heun Functions - 31.12 Confluent Forms of Heun’s Equation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{z}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)\deriv{w}{z}+\frac{\alpha z-q}{z(z-1)}w = 0}
\deriv[2]{w}{z}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)\deriv{w}{z}+\frac{\alpha z-q}{z(z-1)}w = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [z$(2)])+((gamma)/(z)+(delta)/(z - 1)+ epsilon)*diff(w, z)+(alpha*z - q)/(z*(z - 1))*w = 0

D[w, {z, 2}]+(Divide[\[Gamma],z]+Divide[\[Delta],z - 1]+ \[Epsilon])*D[w, z]+Divide[\[Alpha]*z - q,z*(z - 1)]*w == 0
Failure Failure
Failed [300 / 300]
Result: .2500000003-.9330127021*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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, epsilon = 1}


Result: .2500000003-.9330127021*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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, epsilon = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.25, -0.9330127018922194]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϵ, 1]}


Result: Complex[0.25, -0.9330127018922194]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϵ, 2]}

... skip entries to safe data
31.12.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{z}+\left(\frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\deriv{w}{z}+\frac{\alpha z-q}{z^{2}}w = 0}
\deriv[2]{w}{z}+\left(\frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\deriv{w}{z}+\frac{\alpha z-q}{z^{2}}w = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [z$(2)])+((delta)/((z)^(2))+(gamma)/(z)+ 1)*diff(w, z)+(alpha*z - q)/((z)^(2))*w = 0

D[w, {z, 2}]+(Divide[\[Delta],(z)^(2)]+Divide[\[Gamma],z]+ 1)*D[w, z]+Divide[\[Alpha]*z - q,(z)^(2)]*w == 0
Failure Failure
Failed [300 / 300]
Result: .5000000000+0.*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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: .9999999998-1.500000000*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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[0.49999999999999994, -2.7755575615628914*^-17]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.49999999999999994, -2.7755575615628914*^-17]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
31.12.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{z}-\left(\frac{\gamma}{z}+\delta+z\right)\deriv{w}{z}+\frac{\alpha z-q}{z}w = 0}
\deriv[2]{w}{z}-\left(\frac{\gamma}{z}+\delta+z\right)\deriv{w}{z}+\frac{\alpha z-q}{z}w = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [z$(2)])-((gamma)/(z)+ delta + z)*diff(w, z)+(alpha*z - q)/(z)*w = 0

D[w, {z, 2}]-(Divide[\[Gamma],z]+ \[Delta]+ z)*D[w, z]+Divide[\[Alpha]*z - q,z]*w == 0
Failure Failure
Failed [300 / 300]
Result: .4330127020+.2500000000*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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: .7990381058+1.616025404*I
Test Values: {alpha = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, q = 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[0.4330127018922193, 0.24999999999999994]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4330127018922193, 0.24999999999999994]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
31.12.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{w}{z}+\left(\gamma+z\right)z\deriv{w}{z}+\left(\alpha z-q\right)w = 0}
\deriv[2]{w}{z}+\left(\gamma+z\right)z\deriv{w}{z}+\left(\alpha z-q\right)w = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(w, [z$(2)])+(gamma + z)*z*diff(w, z)+(alpha*z - q)*w = 0

D[w, {z, 2}]+(\[Gamma]+ z)*z*D[w, z]+(\[Alpha]*z - q)*w == 0
Failure Failure
Failed [300 / 300]
Result: .2500000002+.4330127020*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, q = 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: -1.799038106-.1160254034*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, q = 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[0.25000000000000006, 0.43301270189221924]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.25000000000000006, 0.43301270189221924]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[γ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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