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 Ξ» = z / a πœ† 𝑧 π‘Ž {\displaystyle{\displaystyle\lambda=z/a}}
\lambda = z/a		 

lambda = z/a
 
\[Lambda] == z/a
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex2 Ξ· = ( 2 ⁒ ( Ξ» - 1 - ln ⁑ Ξ» ) ) 1 / 2 πœ‚ superscript 2 πœ† 1 πœ† 1 2 {\displaystyle{\displaystyle\eta=\left(2(\lambda-1-\ln\lambda)\right)^{1/2}}}
\eta = \left(2(\lambda-1-\ln@@{\lambda})\right)^{1/2}		 

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[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
8.12#Ex3 1 2 ⁒ Ξ· 2 = Ξ» - 1 - ln ⁑ Ξ» 1 2 superscript πœ‚ 2 πœ† 1 πœ† {\displaystyle{\displaystyle\tfrac{1}{2}\eta^{2}=\lambda-1-\ln\lambda}}
\tfrac{1}{2}\eta^{2} = \lambda-1-\ln@@{\lambda}		 

(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[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
8.12#Ex4 d Ξ· d Ξ» = Ξ» - 1 Ξ» ⁒ Ξ· derivative πœ‚ πœ† πœ† 1 πœ† πœ‚ {\displaystyle{\displaystyle\frac{\mathrm{d}\eta}{\mathrm{d}\lambda}=\frac{% \lambda-1}{\lambda\eta}}}
\deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}		 

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[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ», Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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

... skip entries to safe data
8.12.E3 P ⁑ ( a , z ) = 1 2 ⁒ erfc ⁑ ( - Ξ· ⁒ a / 2 ) - S ⁒ ( a , Ξ· ) incomplete-gamma-P π‘Ž 𝑧 1 2 complementary-error-function πœ‚ π‘Ž 2 𝑆 π‘Ž πœ‚ {\displaystyle{\displaystyle P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}% \left(-\eta\sqrt{a/2}\right)-S(a,\eta)}}
\normincGammaP@{a}{z} = \tfrac{1}{2}\erfc@{-\eta\sqrt{a/2}}-S(a,\eta)		 

(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 Q ⁑ ( a , z ) = 1 2 ⁒ erfc ⁑ ( Ξ· ⁒ a / 2 ) + S ⁒ ( a , Ξ· ) incomplete-gamma-Q π‘Ž 𝑧 1 2 complementary-error-function πœ‚ π‘Ž 2 𝑆 π‘Ž πœ‚ {\displaystyle{\displaystyle Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}% \left(\eta\sqrt{a/2}\right)+S(a,\eta)}}
\normincGammaQ@{a}{z} = \tfrac{1}{2}\erfc@{\eta\sqrt{a/2}}+S(a,\eta)		 β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re 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 e + Ο€ ⁒ i ⁒ a 2 ⁒ i ⁒ sin ⁑ ( Ο€ ⁒ a ) ⁒ Q ⁑ ( - a , z ⁒ e + Ο€ ⁒ i ) = + 1 2 ⁒ erfc ⁑ ( + i ⁒ Ξ· ⁒ a / 2 ) - i ⁒ T ⁒ ( a , Ξ· ) superscript 𝑒 πœ‹ 𝑖 π‘Ž 2 𝑖 πœ‹ π‘Ž incomplete-gamma-Q π‘Ž 𝑧 superscript 𝑒 πœ‹ 𝑖 1 2 complementary-error-function 𝑖 πœ‚ π‘Ž 2 𝑖 𝑇 π‘Ž πœ‚ {\displaystyle{\displaystyle\frac{e^{+\pi ia}}{2i\sin\left(\pi a\right)}Q\left% (-a,ze^{+\pi i}\right)=+\tfrac{1}{2}\operatorname{erfc}\left(+i\eta\sqrt{a/2}% \right)-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)		 β„œ ⁑ ( - a ) > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re(-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 e - Ο€ ⁒ i ⁒ a 2 ⁒ i ⁒ sin ⁑ ( Ο€ ⁒ a ) ⁒ Q ⁑ ( - a , z ⁒ e - Ο€ ⁒ i ) = - 1 2 ⁒ erfc ⁑ ( - i ⁒ Ξ· ⁒ a / 2 ) - i ⁒ T ⁒ ( a , Ξ· ) superscript 𝑒 πœ‹ 𝑖 π‘Ž 2 𝑖 πœ‹ π‘Ž incomplete-gamma-Q π‘Ž 𝑧 superscript 𝑒 πœ‹ 𝑖 1 2 complementary-error-function 𝑖 πœ‚ π‘Ž 2 𝑖 𝑇 π‘Ž πœ‚ {\displaystyle{\displaystyle\frac{e^{-\pi ia}}{2i\sin\left(\pi a\right)}Q\left% (-a,ze^{-\pi i}\right)=-\tfrac{1}{2}\operatorname{erfc}\left(-i\eta\sqrt{a/2}% \right)-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)		 β„œ ⁑ ( - a ) > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re(-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 Ξ“ ⁑ ( a + 1 ) ⁒ e + Ο€ ⁒ i ⁒ a 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( - a , z ⁒ e + Ο€ ⁒ i ) = - 1 2 ⁒ erfc ⁑ ( + i ⁒ Ξ· ⁒ a / 2 ) + i ⁒ T ⁒ ( a , Ξ· ) Euler-Gamma π‘Ž 1 superscript 𝑒 πœ‹ 𝑖 π‘Ž 2 πœ‹ 𝑖 incomplete-Gamma π‘Ž 𝑧 superscript 𝑒 πœ‹ 𝑖 1 2 complementary-error-function 𝑖 πœ‚ π‘Ž 2 𝑖 𝑇 π‘Ž πœ‚ {\displaystyle{\displaystyle\Gamma\left(a+1\right)\frac{e^{+\pi ia}}{2\pi i}% \Gamma\left(-a,ze^{+\pi i}\right)=-\tfrac{1}{2}\operatorname{erfc}\left(+i\eta% \sqrt{a/2}\right)+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)		 

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 Ξ“ ⁑ ( a + 1 ) ⁒ e - Ο€ ⁒ i ⁒ a 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( - a , z ⁒ e - Ο€ ⁒ i ) = + 1 2 ⁒ erfc ⁑ ( - i ⁒ Ξ· ⁒ a / 2 ) + i ⁒ T ⁒ ( a , Ξ· ) Euler-Gamma π‘Ž 1 superscript 𝑒 πœ‹ 𝑖 π‘Ž 2 πœ‹ 𝑖 incomplete-Gamma π‘Ž 𝑧 superscript 𝑒 πœ‹ 𝑖 1 2 complementary-error-function 𝑖 πœ‚ π‘Ž 2 𝑖 𝑇 π‘Ž πœ‚ {\displaystyle{\displaystyle\Gamma\left(a+1\right)\frac{e^{-\pi ia}}{2\pi i}% \Gamma\left(-a,ze^{-\pi i}\right)=+\tfrac{1}{2}\operatorname{erfc}\left(-i\eta% \sqrt{a/2}\right)+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)		 

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 z - a ⁒ Ξ³ * ⁑ ( - a , - z ) = cos ⁑ ( Ο€ ⁒ a ) - 2 ⁒ sin ⁑ ( Ο€ ⁒ a ) ⁒ ( e 1 2 ⁒ a ⁒ Ξ· 2 Ο€ ⁒ F ⁑ ( Ξ· ⁒ a / 2 ) + T ⁒ ( a , Ξ· ) ) superscript 𝑧 π‘Ž incomplete-gamma-star π‘Ž 𝑧 πœ‹ π‘Ž 2 πœ‹ π‘Ž superscript 𝑒 1 2 π‘Ž superscript πœ‚ 2 πœ‹ Dawsons-integral πœ‚ π‘Ž 2 𝑇 π‘Ž πœ‚ {\displaystyle{\displaystyle z^{-a}\gamma^{*}\left(-a,-z\right)=\cos\left(\pi a% \right)-2\sin\left(\pi a\right)\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi% }}F\left(\eta\sqrt{a/2}\right)+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)		 β„œ ⁑ ( - a ) > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re(-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 c 0 ⁒ ( Ξ· ) = 1 ΞΌ - 1 Ξ· subscript 𝑐 0 πœ‚ 1 πœ‡ 1 πœ‚ {\displaystyle{\displaystyle c_{0}(\eta)=\frac{1}{\mu}-\frac{1}{\eta}}}
c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}		 

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 c 1 ⁒ ( Ξ· ) = 1 Ξ· 3 - 1 ΞΌ 3 - 1 ΞΌ 2 - 1 12 ⁒ ΞΌ subscript 𝑐 1 πœ‚ 1 superscript πœ‚ 3 1 superscript πœ‡ 3 1 superscript πœ‡ 2 1 12 πœ‡ {\displaystyle{\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}		 

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 c k ⁒ ( Ξ· ) = 1 Ξ· ⁒ d d Ξ· ⁑ c k - 1 ⁒ ( Ξ· ) + ( - 1 ) k ⁒ g k ΞΌ subscript 𝑐 π‘˜ πœ‚ 1 πœ‚ derivative πœ‚ subscript 𝑐 π‘˜ 1 πœ‚ superscript 1 π‘˜ subscript 𝑔 π‘˜ πœ‡ {\displaystyle{\displaystyle c_{k}(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{% \mathrm{d}\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}		 

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[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, 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[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, 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 c k ⁒ ( Ξ· ) = βˆ‘ n = 0 ∞ d k , n ⁒ Ξ· n subscript 𝑐 π‘˜ πœ‚ superscript subscript 𝑛 0 subscript 𝑑 π‘˜ 𝑛 superscript πœ‚ 𝑛 {\displaystyle{\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}		 | Ξ· | < 2 ⁒ Ο€ πœ‚ 2 πœ‹ {\displaystyle{\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 d 0 , n = ( n + 2 ) ⁒ Ξ± n + 2 subscript 𝑑 0 𝑛 𝑛 2 subscript 𝛼 𝑛 2 {\displaystyle{\displaystyle d_{0,n}=(n+2)\alpha_{n+2}}}
d_{0,n} = (n+2)\alpha_{n+2}		 n β‰₯ 1 𝑛 1 {\displaystyle{\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 d k , n = ( - 1 ) k ⁒ g k ⁒ d 0 , n + ( n + 2 ) ⁒ d k - 1 , n + 2 subscript 𝑑 π‘˜ 𝑛 superscript 1 π‘˜ subscript 𝑔 π‘˜ subscript 𝑑 0 𝑛 𝑛 2 subscript 𝑑 π‘˜ 1 𝑛 2 {\displaystyle{\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}		 n β‰₯ 0 , k β‰₯ 1 formulae-sequence 𝑛 0 π‘˜ 1 {\displaystyle{\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 Ξ» - 1 = Ξ· + 1 3 ⁒ Ξ· 2 + βˆ‘ n = 3 ∞ Ξ± n ⁒ Ξ· n πœ† 1 πœ‚ 1 3 superscript πœ‚ 2 superscript subscript 𝑛 3 subscript 𝛼 𝑛 superscript πœ‚ 𝑛 {\displaystyle{\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}		 | Ξ· | < 2 ⁒ Ο€ πœ‚ 2 πœ‹ {\displaystyle{\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 Ξ± 3 = 1 36 subscript 𝛼 3 1 36 {\displaystyle{\displaystyle\alpha_{3}=\tfrac{1}{36}}}
\alpha_{3} = \tfrac{1}{36}		 

alpha[3] = (1)/(36)
 
Subscript[\[Alpha], 3] == Divide[1,36]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex11 Ξ± 4 = - 1 270 subscript 𝛼 4 1 270 {\displaystyle{\displaystyle\alpha_{4}=-\tfrac{1}{270}}}
\alpha_{4} = -\tfrac{1}{270}		 

alpha[4] = -(1)/(270)
 
Subscript[\[Alpha], 4] == -Divide[1,270]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex12 Ξ± 5 = 1 4320 subscript 𝛼 5 1 4320 {\displaystyle{\displaystyle\alpha_{5}=\tfrac{1}{4320}}}
\alpha_{5} = \tfrac{1}{4320}		 

alpha[5] = (1)/(4320)
 
Subscript[\[Alpha], 5] == Divide[1,4320]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex13 Ξ± 6 = 1 17010 subscript 𝛼 6 1 17010 {\displaystyle{\displaystyle\alpha_{6}=\tfrac{1}{17010}}}
\alpha_{6} = \tfrac{1}{17010}		 

alpha[6] = (1)/(17010)
 
Subscript[\[Alpha], 6] == Divide[1,17010]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex14 Ξ± 7 = - 139 54 43200 subscript 𝛼 7 139 54 43200 {\displaystyle{\displaystyle\alpha_{7}=-\tfrac{139}{54\;43200}}}
\alpha_{7} = -\tfrac{139}{54\;43200}		 

alpha[7] = -(139)/(5443200)
 
Subscript[\[Alpha], 7] == -Divide[139,5443200]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex15 Ξ± 8 = 1 2 04120 subscript 𝛼 8 1 2 04120 {\displaystyle{\displaystyle\alpha_{8}=\tfrac{1}{2\;04120}}}
\alpha_{8} = \tfrac{1}{2\;04120}		 

alpha[8] = (1)/(204120)
 
Subscript[\[Alpha], 8] == Divide[1,204120]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex16 c 0 ⁒ ( 0 ) = - 1 3 subscript 𝑐 0 0 1 3 {\displaystyle{\displaystyle c_{0}(0)=-\tfrac{1}{3}}}
c_{0}(0) = -\tfrac{1}{3}		 

c[0](0) = -(1)/(3)
 
Subscript[c, 0][0] == -Divide[1,3]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex17 c 1 ⁒ ( 0 ) = - 1 540 subscript 𝑐 1 0 1 540 {\displaystyle{\displaystyle c_{1}(0)=-\tfrac{1}{540}}}
c_{1}(0) = -\tfrac{1}{540}		 

c[1](0) = -(1)/(540)
 
Subscript[c, 1][0] == -Divide[1,540]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex18 c 2 ⁒ ( 0 ) = 25 6048 subscript 𝑐 2 0 25 6048 {\displaystyle{\displaystyle c_{2}(0)=\tfrac{25}{6048}}}
c_{2}(0) = \tfrac{25}{6048}		 

c[2](0) = (25)/(6048)
 
Subscript[c, 2][0] == Divide[25,6048]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex19 c 3 ⁒ ( 0 ) = 101 1 55520 subscript 𝑐 3 0 101 1 55520 {\displaystyle{\displaystyle c_{3}(0)=\tfrac{101}{1\;55520}}}
c_{3}(0) = \tfrac{101}{1\;55520}		 

c[3](0) = (101)/(155520)
 
Subscript[c, 3][0] == Divide[101,155520]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex20 c 4 ⁒ ( 0 ) = - 31 84811 36951 55200 subscript 𝑐 4 0 31 84811 36951 55200 {\displaystyle{\displaystyle c_{4}(0)=-\tfrac{31\;84811}{36951\;55200}}}
c_{4}(0) = -\tfrac{31\;84811}{36951\;55200}		 

c[4](0) = -(3184811)/(3695155200)
 
Subscript[c, 4][0] == -Divide[3184811,3695155200]
 Skipped - no semantic math Skipped - no semantic math - -
8.12#Ex21 c 5 ⁒ ( 0 ) = - 27 45493 81517 36320 subscript 𝑐 5 0 27 45493 81517 36320 {\displaystyle{\displaystyle c_{5}(0)=-\tfrac{27\;45493}{81517\;36320}}}
c_{5}(0) = -\tfrac{27\;45493}{81517\;36320}		 

c[5](0) = -(2745493)/(8151736320)
 
Subscript[c, 5][0] == -Divide[2745493,8151736320]
 Skipped - no semantic math Skipped - no semantic math - -
8.12.E21 Q ⁑ ( a , x ) = q incomplete-gamma-Q π‘Ž π‘₯ π‘ž {\displaystyle{\displaystyle Q\left(a,x\right)=q}}
\normincGammaQ@{a}{x} = q		 β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re 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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