Exponential, Logarithmic, Sine, and Cosine Integrals - 6.6 Power Series

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
6.6.E1	${\displaystyle{\displaystyle\mathrm{Ei}\left(x\right)=\gamma+\ln x+\sum_{n=1}^{\infty}\frac{x^{n}}{n!\thinspace n}}}$ \expintEi@{x} = \EulerConstant+\ln@@{x}+\sum_{n=1}^{\infty}\frac{x^{n}}{n!\thinspace n}	${\displaystyle{\displaystyle x>0}}$	Error	ExpIntegralEi[x] == EulerGamma + Log[x]+ Sum[Divide[(x)^(n),(n)!*n], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Successful [Tested: 3]
6.6.E2	${\displaystyle{\displaystyle E_{1}\left(z\right)=-\gamma-\ln z-\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{n}}{n!\thinspace n}}}$ \expintE@{z} = -\EulerConstant-\ln@@{z}-\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{n}}{n!\thinspace n}		Ei(z) = - gamma - ln(z)- sum(((- 1)^(n)* (z)^(n))/(factorial(n)*n), n = 1..infinity)	ExpIntegralE[1, z] == - EulerGamma - Log[z]- Sum[Divide[(- 1)^(n)* (z)^(n),(n)!*n], {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [7 / 7] Result: 1.393548628+1.498247032*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .8944744989+3.773814377*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
6.6.E3	${\displaystyle{\displaystyle E_{1}\left(z\right)=-\ln z+e^{-z}\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\psi\left(n+1\right)}}$ \expintE@{z} = -\ln@@{z}+e^{-z}\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\digamma@{n+1}		Ei(z) = - ln(z)+ exp(- z)*sum(((z)^(n))/(factorial(n))*Psi(n + 1), n = 0..infinity)	ExpIntegralE[1, z] == - Log[z]+ Exp[- z]*Sum[Divide[(z)^(n),(n)!]*PolyGamma[n + 1], {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [7 / 7] Result: 1.393548628+1.498247031*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .8944744987+3.773814376*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
6.6.E4	${\displaystyle{\displaystyle\mathrm{Ein}\left(z\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^{n}}{n!\thinspace n}}}$ \expintEin@{z} = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^{n}}{n!\thinspace n}		Error	ExpIntegralE[1, z] + Ln[z] + EulerGamma == Sum[Divide[(- 1)^(n - 1)* (z)^(n),(n)!*n], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[0.0, -0.5235987755982988], Ln[Complex[0.8660254037844387, 0.49999999999999994]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.0, -2.0943951023931953], Ln[Complex[-0.4999999999999998, 0.8660254037844387]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
6.6.E5	${\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{(2n+1)!(2n+1)}}}$ \sinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{(2n+1)!(2n+1)}		Si(z) = sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(2*n + 1)*(2*n + 1)), n = 0..infinity)	SinIntegral[z] == Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(2*n + 1)!*(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
6.6.E6	${\displaystyle{\displaystyle\mathrm{Ci}\left(z\right)=\gamma+\ln z+\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{2n}}{(2n)!(2n)}}}$ \cosint@{z} = \EulerConstant+\ln@@{z}+\sum_{n=1}^{\infty}\frac{(-1)^{n}z^{2n}}{(2n)!(2n)}		Ci(z) = gamma + ln(z)+ sum(((- 1)^(n)* (z)^(2*n))/(factorial(2*n)*(2*n)), n = 1..infinity)	CosIntegral[z] == EulerGamma + Log[z]+ Sum[Divide[(- 1)^(n)* (z)^(2*n),(2*n)!*(2*n)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]