Incomplete Gamma and Related Functions - 8.12 Uniform Asymptotic Expansions for Large Parameter

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.12#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = z/a}
\lambda = z/a		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
lambda = z/a
 
\[Lambda] == z/a
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \eta = \left(2(\lambda-1-\ln@@{\lambda})\right)^{1/2}}
\eta = \left(2(\lambda-1-\ln@@{\lambda})\right)^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
eta = (2*(lambda - 1 - ln(lambda)))^(1/2)

\[Eta] == (2*(\[Lambda]- 1 - Log[\[Lambda]]))^(1/2)
Failure Failure
Failed [100 / 100]
Result: .8206105237+1.019626504*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}


Result: .2036159778+2.354396465*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [100 / 100]
Result: Complex[0.8206105232686428, 1.019626504138681]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.20361597732323333, 2.3543964646926674]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.12#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tfrac{1}{2}\eta^{2} = \lambda-1-\ln@@{\lambda}}
\tfrac{1}{2}\eta^{2} = \lambda-1-\ln@@{\lambda}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(1)/(2)*(eta)^(2) = lambda - 1 - ln(lambda)

Divide[1,2]*\[Eta]^(2) == \[Lambda]- 1 - Log[\[Lambda]]
Failure Failure
Failed [100 / 100]
Result: .3839745964+.4566114775*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}


Result: 1.750000000+1.661382400*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [100 / 100]
Result: Complex[0.38397459621556135, 0.45661147749051817]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.7499999999999998, 1.6613824005009756]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.12#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}}
\deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(eta, lambda) = (lambda - 1)/(lambda*eta)

D[\[Eta], \[Lambda]] == Divide[\[Lambda]- 1,\[Lambda]*\[Eta]]
Failure Failure
Failed [100 / 100]
Result: -.3660254037-.3660254035*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}


Result: -1.732050807-.2266367838e-9*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [100 / 100]
Result: Complex[-0.3660254037844386, -0.3660254037844386]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.7320508075688772, 3.3306690738754696*^-16]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.12.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \normincGammaP@{a}{z} = \tfrac{1}{2}\erfc@{-\eta\sqrt{a/2}}-S(a,\eta)}
\normincGammaP@{a}{z} = \tfrac{1}{2}\erfc@{-\eta\sqrt{a/2}}-S(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(2)*erfc(- eta*sqrt(a/2))- S(a , eta)

GammaRegularized[a, 0, z] == Divide[1,2]*Erfc[- \[Eta]*Sqrt[a/2]]- S[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: .8724483635-.3325384943*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .8436948583-.7685914925*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \normincGammaQ@{a}{z} = \tfrac{1}{2}\erfc@{\eta\sqrt{a/2}}+S(a,\eta)}
\normincGammaQ@{a}{z} = \tfrac{1}{2}\erfc@{\eta\sqrt{a/2}}+S(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
GAMMA(a, z)/GAMMA(a) = (1)/(2)*erfc(eta*sqrt(a/2))+ S(a , eta)

GammaRegularized[a, z] == Divide[1,2]*Erfc[\[Eta]*Sqrt[a/2]]+ S[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: -.8724483631+.3325384943*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.8436948582+.7685914925*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{e^{+\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{+\pi i}} = +\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}-iT(a,\eta)}
\frac{e^{+\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{+\pi i}} = +\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}-iT(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(-a)} > 0} 
(exp(+ Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(+ Pi*I))/GAMMA(- a) = +(1)/(2)*erfc(+ I*eta*sqrt(a/2))- I*T(a , eta)

Divide[Exp[+ Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[+ Pi*I]] == +Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/2]]- I*T[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: .1738836865-.4215091763*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.5322485765+.1038051776*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{e^{-\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{-\pi i}} = -\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}-iT(a,\eta)}
\frac{e^{-\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{-\pi i}} = -\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}-iT(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(-a)} > 0} 
(exp(- Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(- Pi*I))/GAMMA(- a) = -(1)/(2)*erfc(- I*eta*sqrt(a/2))- I*T(a , eta)

Divide[Exp[- Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[- Pi*I]] == -Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/2]]- I*T[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: -.9809290254+.1461521622*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.2747967621-.3791621909*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{a+1}\frac{e^{+\pi ia}}{2\pi i}\incGamma@{-a}{ze^{+\pi i}} = -\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}+iT(a,\eta)}
\EulerGamma@{a+1}\frac{e^{+\pi ia}}{2\pi i}\incGamma@{-a}{ze^{+\pi i}} = -\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}+iT(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
GAMMA(a + 1)*(exp(+ Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(+ Pi*I)) = -(1)/(2)*erfc(+ I*eta*sqrt(a/2))+ I*T(a , eta)

Gamma[a + 1]*Divide[Exp[+ Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[+ Pi*I]] == -Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/2]]+ I*T[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: -.1738836865+.4215091762*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .5322485766-.1038051776*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{a+1}\frac{e^{-\pi ia}}{2\pi i}\incGamma@{-a}{ze^{-\pi i}} = +\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}+iT(a,\eta)}
\EulerGamma@{a+1}\frac{e^{-\pi ia}}{2\pi i}\incGamma@{-a}{ze^{-\pi i}} = +\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}+iT(a,\eta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
GAMMA(a + 1)*(exp(- Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(- Pi*I)) = +(1)/(2)*erfc(- I*eta*sqrt(a/2))+ I*T(a , eta)

Gamma[a + 1]*Divide[Exp[- Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[- Pi*I]] == +Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/2]]+ I*T[a , \[Eta]]
Failure Failure
Failed [300 / 300]
Result: .9809290254-.1461521621*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: .2747967620+.3791621907*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
8.12.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z^{-a}\scincgamma@{-a}{-z} = \cos@{\pi a}-2\sin@{\pi a}\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}\DawsonsintF@{\eta\sqrt{a/2}}+T(a,\eta)\right)}
z^{-a}\scincgamma@{-a}{-z} = \cos@{\pi a}-2\sin@{\pi a}\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}\DawsonsintF@{\eta\sqrt{a/2}}+T(a,\eta)\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(-a)} > 0} 
(z)^(- a)* (- z)^(-(- a))*(GAMMA(- a)-GAMMA(- a, - z))/GAMMA(- a) = cos(Pi*a)- 2*sin(Pi*a)*((exp((1)/(2)*a*(eta)^(2)))/(sqrt(Pi))*dawson(eta*sqrt(a/2))+ T(a , eta))

Error
Failure Missing Macro Error
Failed [300 / 300]
Result: .2923043261+1.961858052*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -.7583243808+.5495935246*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 -
8.12#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}}
c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[0](eta) = (1)/(mu)-(1)/(eta)
 
Subscript[c, 0][\[Eta]] == Divide[1,\[Mu]]-Divide[1,\[Eta]]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{1}(\eta) = \frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{12\mu}}
c_{1}(\eta) = \frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{12\mu}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[1](eta) = (1)/((eta)^(3))-(1)/((mu)^(3))-(1)/((mu)^(2))-(1)/(12*mu)
 
Subscript[c, 1][\[Eta]] == Divide[1,\[Eta]^(3)]-Divide[1,\[Mu]^(3)]-Divide[1,\[Mu]^(2)]-Divide[1,12*\[Mu]]
 Skipped - no semantic math Skipped - no semantic math - -
8.12.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{k}(\eta) = \frac{1}{\eta}\deriv{}{\eta}c_{k-1}(\eta)+(-1)^{k}\frac{g_{k}}{\mu}}
c_{k}(\eta) = \frac{1}{\eta}\deriv{}{\eta}c_{k-1}(\eta)+(-1)^{k}\frac{g_{k}}{\mu}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[k](eta) = (1)/(eta)*diff(c[k - 1](eta), eta)+(- 1)^(k)*(g[k])/(mu)

Subscript[c, k][\[Eta]] == Divide[1,\[Eta]]*D[Subscript[c, k - 1][\[Eta]], \[Eta]]+(- 1)^(k)*Divide[Subscript[g, k],\[Mu]]
Failure Failure
Failed [300 / 300]
Result: .5000000004+.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I, c[k-1] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}


Result: -1.500000000+.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I, c[k-1] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[k, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, Plus[-1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.5, 0.8660254037844386]
Test Values: {Rule[k, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, Plus[-1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.12.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{k}(\eta) = \sum_{n=0}^{\infty}d_{k,n}\eta^{n}}
c_{k}(\eta) = \sum_{n=0}^{\infty}d_{k,n}\eta^{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\eta| < 2\sqrt{\pi}} 
c[k](eta) = sum(d[k , n]*(eta)^(n), n = 0..infinity)
 
Subscript[c, k][\[Eta]] == Sum[Subscript[d, k , n]*\[Eta]^(n), {n, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d_{0,n} = (n+2)\alpha_{n+2}}
d_{0,n} = (n+2)\alpha_{n+2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 1} 
d[0 , n] = (n + 2)*alpha[n + 2]
 
Subscript[d, 0 , n] == (n + 2)*Subscript[\[Alpha], n + 2]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d_{k,n} = (-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2}}
d_{k,n} = (-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n \geq 0, k \geq 1} 
d[k , n] = (- 1)^(k)* g[k]*d[0 , n]+(n + 2)*d[k - 1 , n + 2]
 
Subscript[d, k , n] == (- 1)^(k)* Subscript[g, k]*Subscript[d, 0 , n]+(n + 2)*Subscript[d, k - 1 , n + 2]
 Skipped - no semantic math Skipped - no semantic math - -
8.12.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda-1 = \eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n}}
\lambda-1 = \eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\eta| < 2\sqrt{\pi}} 
lambda - 1 = eta +(1)/(3)*(eta)^(2)+ sum(alpha[n]*(eta)^(n), n = 3..infinity)
 
\[Lambda]- 1 == \[Eta]+Divide[1,3]*\[Eta]^(2)+ Sum[Subscript[\[Alpha], n]*\[Eta]^(n), {n, 3, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{3} = \tfrac{1}{36}}
\alpha_{3} = \tfrac{1}{36}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[3] = (1)/(36)
 
Subscript[\[Alpha], 3] == Divide[1,36]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{4} = -\tfrac{1}{270}}
\alpha_{4} = -\tfrac{1}{270}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[4] = -(1)/(270)
 
Subscript[\[Alpha], 4] == -Divide[1,270]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{5} = \tfrac{1}{4320}}
\alpha_{5} = \tfrac{1}{4320}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[5] = (1)/(4320)
 
Subscript[\[Alpha], 5] == Divide[1,4320]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{6} = \tfrac{1}{17010}}
\alpha_{6} = \tfrac{1}{17010}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[6] = (1)/(17010)
 
Subscript[\[Alpha], 6] == Divide[1,17010]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{7} = -\tfrac{139}{54\;43200}}
\alpha_{7} = -\tfrac{139}{54\;43200}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[7] = -(139)/(5443200)
 
Subscript[\[Alpha], 7] == -Divide[139,5443200]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{8} = \tfrac{1}{2\;04120}}
\alpha_{8} = \tfrac{1}{2\;04120}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[8] = (1)/(204120)
 
Subscript[\[Alpha], 8] == Divide[1,204120]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{0}(0) = -\tfrac{1}{3}}
c_{0}(0) = -\tfrac{1}{3}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[0](0) = -(1)/(3)
 
Subscript[c, 0][0] == -Divide[1,3]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{1}(0) = -\tfrac{1}{540}}
c_{1}(0) = -\tfrac{1}{540}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[1](0) = -(1)/(540)
 
Subscript[c, 1][0] == -Divide[1,540]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{2}(0) = \tfrac{25}{6048}}
c_{2}(0) = \tfrac{25}{6048}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[2](0) = (25)/(6048)
 
Subscript[c, 2][0] == Divide[25,6048]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{3}(0) = \tfrac{101}{1\;55520}}
c_{3}(0) = \tfrac{101}{1\;55520}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[3](0) = (101)/(155520)
 
Subscript[c, 3][0] == Divide[101,155520]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex20 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{4}(0) = -\tfrac{31\;84811}{36951\;55200}}
c_{4}(0) = -\tfrac{31\;84811}{36951\;55200}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[4](0) = -(3184811)/(3695155200)
 
Subscript[c, 4][0] == -Divide[3184811,3695155200]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex21 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{5}(0) = -\tfrac{27\;45493}{81517\;36320}}
c_{5}(0) = -\tfrac{27\;45493}{81517\;36320}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c[5](0) = -(2745493)/(8151736320)
 
Subscript[c, 5][0] == -Divide[2745493,8151736320]
 Skipped - no semantic math Skipped - no semantic math - -
8.12.E21 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \normincGammaQ@{a}{x} = q}
\normincGammaQ@{a}{x} = q		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
GAMMA(a, x)/GAMMA(a) = q

GammaRegularized[a, x] == q
Failure Failure
Failed [180 / 180]
Result: -.8512854784-.5000000000*I
Test Values: {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: -.5487148961-.5000000000*I
Test Values: {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = .5}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[-0.8512854781447857, -0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[-0.5487148959215247, -0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
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