Incomplete Gamma and Related Functions - 8.11 Asymptotic Approximations and Expansions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.11.E2	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)}}$ \incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)		GAMMA(a, z) = (z)^(a - 1)* exp(- z)*(sum(((- 1)^(k)* pochhammer(1 - a, k))/((z)^(k)), k = 0..n - 1)+ R[n](a , z))	Gamma[a, z] == (z)^(a - 1)* Exp[- z]*(Sum[Divide[(- 1)^(k)* Pochhammer[1 - a, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[R, n][a , z])	Failure	Failure	Failed [300 / 300] Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(1.000000000+0.*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)) Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 1, n = 3} Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(-1.165063509+1.250000000*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)) Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 2, n = 3} ... skip entries to safe data	Error
8.11.E4	${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{{\left(a\right)_{k+1}}}}}$ \incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}	${\displaystyle{\displaystyle\Re a>0}}$	GAMMA(a)-GAMMA(a, z) = (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)	Gamma[a, 0, z] == (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
8.11#Ex1	${\displaystyle{\displaystyle b_{0}(\lambda)=1}}$ b_{0}(\lambda) = 1		b[0](lambda) = 1	Subscript[b, 0][\[Lambda]] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
8.11#Ex2	${\displaystyle{\displaystyle b_{1}(\lambda)=\lambda}}$ b_{1}(\lambda) = \lambda		b[1](lambda) = lambda	Subscript[b, 1][\[Lambda]] == \[Lambda]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.11#Ex3	${\displaystyle{\displaystyle b_{2}(\lambda)=\lambda(2\lambda+1)}}$ b_{2}(\lambda) = \lambda(2\lambda+1)		b[2](lambda) = lambda*(2*lambda + 1)	Subscript[b, 2][\[Lambda]] == \[Lambda]*(2*\[Lambda]+ 1)	Skipped - no semantic math	Skipped - no semantic math	-	-
8.11.E15	${\displaystyle{\displaystyle S_{n}(x)=\frac{\gamma\left(n+1,nx\right)}{(nx)^{n}e^{-nx}}}}$ S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}	${\displaystyle{\displaystyle\Re(n+1)>0}}$	S[n](x) = (GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x))	Subscript[S, n][x] == Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]	Failure	Failure	Failed [90 / 90] Result: -.22087941e-1+.7500000000*I Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -1.275525655+.7500000000*I Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[-0.02208794121538471, 0.7499999999999999] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.2755256550317124, 0.7499999999999999] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.11.E19	${\displaystyle{\displaystyle d_{k}(x)=\frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}}}$ d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}		d[k](x) = ((- 1)^(k)* b[k](x))/((1 - x)^(2*k + 1))	Subscript[d, k][x] == Divide[(- 1)^(k)* Subscript[b, k][x],(1 - x)^(2*k + 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-