Incomplete Gamma and Related Functions - 8.20 Asymptotic Expansions of

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.20.E1	${\displaystyle{\displaystyle E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left(p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p}\left(z\right)\right)}}$ \genexpintE{p}@{z} = \frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{\Pochhammersym{p}{k}}{z^{k}}+(-1)^{n}\frac{\Pochhammersym{p}{n}e^{z}}{z^{n-1}}\genexpintE{n+p}@{z}\right)		Ei(p, z) = (exp(- z))/(z)*(sum((- 1)^(k)*(pochhammer(p, k))/((z)^(k)), k = 0..n - 1)+(- 1)^(n)*(pochhammer(p, n)*exp(z))/((z)^(n - 1))*Ei(n + p, z))	ExpIntegralE[p, z] == Divide[Exp[- z],z]*(Sum[(- 1)^(k)*Divide[Pochhammer[p, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n)*Divide[Pochhammer[p, n]*Exp[z],(z)^(n - 1)]*ExpIntegralE[n + p, z])	Failure	Successful	Manual Skip!	Failed [7 / 70] Result: Indeterminate Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.20.E4	${\displaystyle{\displaystyle A_{k+1}(\lambda)=(1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\frac{\mathrm{d}A_{k}(\lambda)}{\mathrm{d}\lambda}}}$ A_{k+1}(\lambda) = (1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\deriv{A_{k}(\lambda)}{\lambda}		A[k + 1](lambda) = (1 - 2*k*lambda)*A[k](lambda)+ lambda*(lambda + 1)*diff(A[k](lambda), lambda)	Subscript[A, k + 1][\[Lambda]] == (1 - 2*k*\[Lambda])*Subscript[A, k][\[Lambda]]+ \[Lambda]*(\[Lambda]+ 1)*D[Subscript[A, k][\[Lambda]], \[Lambda]]	Failure	Failure	Failed [300 / 300] Result: -.5000000000+.133974596*I Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 1, k = 3} Result: -.4999999993+2.133974597*I Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 2, k = 3} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.49999999999999944, 4.133974596215561] Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.8660254037844384, 3.7679491924311224] Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.20#Ex1	${\displaystyle{\displaystyle A_{1}(\lambda)=1}}$ A_{1}(\lambda) = 1		A[1](lambda) = 1	Subscript[A, 1][\[Lambda]] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
8.20#Ex2	${\displaystyle{\displaystyle A_{2}(\lambda)=1-2\lambda}}$ A_{2}(\lambda) = 1-2\lambda		A[2](lambda) = 1 - 2*lambda	Subscript[A, 2][\[Lambda]] == 1 - 2*\[Lambda]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.20#Ex3	${\displaystyle{\displaystyle A_{3}(\lambda)=1-8\lambda+6\lambda^{2}}}$ A_{3}(\lambda) = 1-8\lambda+6\lambda^{2}		A[3](lambda) = 1 - 8*lambda + 6*(lambda)^(2)	Subscript[A, 3][\[Lambda]] == 1 - 8*\[Lambda]+ 6*\[Lambda]^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-