8.19

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.19.E1	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z\right)}}$ \genexpintE{p}@{z} = z^{p-1}\incGamma@{1-p}{z}		Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p, z)	ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p, z]	Successful	Successful	-	Successful [Tested: 70]
8.19.E2	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\mathrm{d}t}}$ \genexpintE{p}@{z} = z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\diff{t}		Ei(p, z) = (z)^(p - 1)* int((exp(- t))/((t)^(p)), t = z..infinity)	ExpIntegralE[p, z] == (z)^(p - 1)* Integrate[Divide[Exp[- t],(t)^(p)], {t, z, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 70]
8.19.E3	${\displaystyle{\displaystyle E_{p}\left(z\right)=\int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\mathrm{d}t}}$ \genexpintE{p}@{z} = \int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$	Ei(p, z) = int((exp(- z*t))/((t)^(p)), t = 1..infinity)	ExpIntegralE[p, z] == Integrate[Divide[Exp[- z*t],(t)^(p)], {t, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 50]
8.19.E4	${\displaystyle{\displaystyle E_{p}\left(z\right)=\frac{z^{p-1}e^{-z}}{\Gamma\left(p\right)}\int_{0}^{\infty}\frac{t^{p-1}e^{-zt}}{1+t}\mathrm{d}t}}$ \genexpintE{p}@{z} = \frac{z^{p-1}e^{-z}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{t^{p-1}e^{-zt}}{1+t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re p>0}}$	Ei(p, z) = ((z)^(p - 1)* exp(- z))/(GAMMA(p))*int(((t)^(p - 1)* exp(- z*t))/(1 + t), t = 0..infinity)	ExpIntegralE[p, z] == Divide[(z)^(p - 1)* Exp[- z],Gamma[p]]*Integrate[Divide[(t)^(p - 1)* Exp[- z*t],1 + t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 25]
8.19.E5	${\displaystyle{\displaystyle E_{0}\left(z\right)=z^{-1}e^{-z}}}$ \genexpintE{0}@{z} = z^{-1}e^{-z}	${\displaystyle{\displaystyle z\neq 0}}$	Ei(0, z) = (z)^(- 1)* exp(- z)	ExpIntegralE[0, z] == (z)^(- 1)* Exp[- z]	Successful	Successful	-	Successful [Tested: 7]
8.19.E6	${\displaystyle{\displaystyle E_{p}\left(0\right)=\frac{1}{p-1}}}$ \genexpintE{p}@{0} = \frac{1}{p-1}	${\displaystyle{\displaystyle\Re p>1}}$	Ei(p, 0) = (1)/(p - 1)	ExpIntegralE[p, 0] == Divide[1,p - 1]	Successful	Successful	-	Successful [Tested: 2]
8.19.E7	${\displaystyle{\displaystyle E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}E_{1}\left(z\right)+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}}}$ \genexpintE{n}@{z} = \frac{(-z)^{n-1}}{(n-1)!}\expintE@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}		Ei(n, z) = ((- z)^(n - 1))/(factorial(n - 1))*Ei(z)+(exp(- z))/(factorial(n - 1))*sum(factorial(n - k - 2)*(- z)^(k), k = 0..n - 2)	ExpIntegralE[n, z] == Divide[(- z)^(n - 1),(n - 1)!]*ExpIntegralE[1, z]+Divide[Exp[- z],(n - 1)!]*Sum[(n - k - 2)!*(- z)^(k), {k, 0, n - 2}, GenerateConditions->None]	Failure	Failure	Failed [21 / 21] Result: -1.393548628-1.498247032*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 1, n = 2} Result: .4577249979+1.994294304*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 2, n = 2} ... skip entries to safe data	Successful [Tested: 21]
8.19.E9	${\displaystyle{\displaystyle E_{n}\left(z\right)=\frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln z+\frac{e^{-z}}{(n-1)!}\sum_{k=1}^{n-1}(-z)^{k-1}\Gamma\left(n-k\right)+\frac{e^{-z}(-z)^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\psi\left(k+1\right)}}$ \genexpintE{n}@{z} = \frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln@@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=1}^{n-1}(-z)^{k-1}\EulerGamma@{n-k}+\frac{e^{-z}(-z)^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\digamma@{k+1}	${\displaystyle{\displaystyle\Re(n-k)>0}}$	Ei(n, z) = ((- 1)^(n)* (z)^(n - 1))/(factorial(n - 1))*ln(z)+(exp(- z))/(factorial(n - 1))*sum((- z)^(k - 1)* GAMMA(n - k), k = 1..n - 1)+(exp(- z)*(- z)^(n - 1))/(factorial(n - 1))*sum(((z)^(k))/(factorial(k))*Psi(k + 1), k = 0..infinity)	ExpIntegralE[n, z] == Divide[(- 1)^(n)* (z)^(n - 1),(n - 1)!]*Log[z]+Divide[Exp[- z],(n - 1)!]*Sum[(- z)^(k - 1)* Gamma[n - k], {k, 1, n - 1}, GenerateConditions->None]+Divide[Exp[- z]*(- z)^(n - 1),(n - 1)!]*Sum[Divide[(z)^(k),(k)!]*PolyGamma[k + 1], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 21]	Failed [16 / 21] Result: Plus[Complex[0.3691288000469654, -0.2016559825387078], Times[Complex[0.2188469268397846, -0.35920360372711485], Plus[-1.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Power[-1, Rational[1, 6]], []], Times[Plus[1, Times[-1, Power[-1, Rational[1, 6]]], Times[-1, ]], [Plus[1, ]]], Times[Plus[-1, ], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], 1]}]][2.0]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.0256870546657635, -0.10579058942927923], Times[Complex[0.1094234634198923, -0.17960180186355743], Plus[-2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Power[-1, Rational[1, 6]], []], Times[Plus[2, Times[-1, Power[-1, Rational[1, 6]]], Times[-1, ]], [Plus[1, ]]], Times[Plus[-2, ], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], 2]}]][3.0]]]], {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.19.E10	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p\right)-\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(1-p+k)}}}$ \genexpintE{p}@{z} = z^{p-1}\EulerGamma@{1-p}-\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(1-p+k)}	${\displaystyle{\displaystyle\Re(1-p)>0}}$	Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p)- sum(((- z)^(k))/(factorial(k)*(1 - p + k)), k = 0..infinity)	ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p]- Sum[Divide[(- z)^(k),(k)!*(1 - p + k)], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 56]
8.19.E11	${\displaystyle{\displaystyle E_{p}\left(z\right)=\Gamma\left(1-p\right)\left(z^{p-1}-e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(2-p+k\right)}\right)}}$ \genexpintE{p}@{z} = \EulerGamma@{1-p}\left(z^{p-1}-e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{2-p+k}}\right)	${\displaystyle{\displaystyle\Re(1-p)>0,\Re(2-p+k)>0}}$	Ei(p, z) = GAMMA(1 - p)*((z)^(p - 1)- exp(- z)*sum(((z)^(k))/(GAMMA(2 - p + k)), k = 0..infinity))	ExpIntegralE[p, z] == Gamma[1 - p]*((z)^(p - 1)- Exp[- z]*Sum[Divide[(z)^(k),Gamma[2 - p + k]], {k, 0, Infinity}, GenerateConditions->None])	Successful	Successful	-	Successful [Tested: 56]
8.19.E12	${\displaystyle{\displaystyle pE_{p+1}\left(z\right)+zE_{p}\left(z\right)=e^{-z}}}$ p\genexpintE{p+1}@{z}+z\genexpintE{p}@{z} = e^{-z}		p*Ei(p + 1, z)+ z*Ei(p, z) = exp(- z)	p*ExpIntegralE[p + 1, z]+ z*ExpIntegralE[p, z] == Exp[- z]	Successful	Successful	-	Successful [Tested: 70]
8.19.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}E_{p}\left(z\right)=-E_{p-1}\left(z\right)}}$ \deriv{}{z}\genexpintE{p}@{z} = -\genexpintE{p-1}@{z}		diff(Ei(p, z), z) = - Ei(p - 1, z)	D[ExpIntegralE[p, z], z] == - ExpIntegralE[p - 1, z]	Successful	Successful	-	Successful [Tested: 70]
8.19.E14	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(e^{z}E_{p}\left(z\right))=e^{z}E_{p}\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}}}$ \deriv{}{z}(e^{z}\genexpintE{p}@{z}) = e^{z}\genexpintE{p}@{z}\left(1+\frac{p-1}{z}\right)-\frac{1}{z}		diff(exp(z)*Ei(p, z), z) = exp(z)*Ei(p, z)*(1 +(p - 1)/(z))-(1)/(z)	D[Exp[z]*ExpIntegralE[p, z], z] == Exp[z]*ExpIntegralE[p, z]*(1 +Divide[p - 1,z])-Divide[1,z]	Successful	Successful	-	Successful [Tested: 70]
8.19.E15	${\displaystyle{\displaystyle\frac{{\partial}^{j}E_{p}\left(z\right)}{{\partial p}^{j}}=(-1)^{j}\int_{1}^{\infty}(\ln t)^{j}t^{-p}e^{-zt}\mathrm{d}t}}$ \pderiv[j]{\genexpintE{p}@{z}}{p} = (-1)^{j}\int_{1}^{\infty}(\ln@@{t})^{j}t^{-p}e^{-zt}\diff{t}	${\displaystyle{\displaystyle\Re z>0}}$	diff(Ei(p, z), [p$(j)]) = (- 1)^(j)* int((ln(t))^(j)* (t)^(- p)* exp(- z*t), t = 1..infinity)	D[ExpIntegralE[p, z], {p, j}] == (- 1)^(j)* Integrate[(Log[t])^(j)* (t)^(- p)* Exp[- z*t], {t, 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
8.19.E16	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}e^{-z}U\left(p,p,z\right)}}$ \genexpintE{p}@{z} = z^{p-1}e^{-z}\KummerconfhyperU@{p}{p}{z}		Ei(p, z) = (z)^(p - 1)* exp(- z)*KummerU(p, p, z)	ExpIntegralE[p, z] == (z)^(p - 1)* Exp[- z]*HypergeometricU[p, p, z]	Successful	Successful	-	Successful [Tested: 70]
8.19.E19	${\displaystyle{\displaystyle\frac{n-1}{n}E_{n}\left(x\right)<E_{n+1}\left(x\right)}}$ \frac{n-1}{n}\genexpintE{n}@{x} < \genexpintE{n+1}@{x}		(n - 1)/(n)*Ei(n, x) < Ei(n + 1, x)	Divide[n - 1,n]*ExpIntegralE[n, x] < ExpIntegralE[n + 1, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E19	${\displaystyle{\displaystyle E_{n+1}\left(x\right)<E_{n}\left(x\right)}}$ \genexpintE{n+1}@{x} < \genexpintE{n}@{x}		Ei(n + 1, x) < Ei(n, x)	ExpIntegralE[n + 1, x] < ExpIntegralE[n, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E20	${\displaystyle{\displaystyle\left(E_{n}\left(x\right)\right)^{2}<E_{n-1}\left(x\right)E_{n+1}\left(x\right)}}$ \left(\genexpintE{n}@{x}\right)^{2} < \genexpintE{n-1}@{x}\genexpintE{n+1}@{x}		(Ei(n, x))^(2) < Ei(n - 1, x)*Ei(n + 1, x)	(ExpIntegralE[n, x])^(2) < ExpIntegralE[n - 1, x]*ExpIntegralE[n + 1, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E21	${\displaystyle{\displaystyle\frac{1}{x+n}<e^{x}E_{n}\left(x\right)}}$ \frac{1}{x+n} < e^{x}\genexpintE{n}@{x}		(1)/(x + n) < exp(x)*Ei(n, x)	Divide[1,x + n] < Exp[x]*ExpIntegralE[n, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E21	${\displaystyle{\displaystyle e^{x}E_{n}\left(x\right)\leq\frac{1}{x+n-1}}}$ e^{x}\genexpintE{n}@{x} \leq \frac{1}{x+n-1}		exp(x)*Ei(n, x) <= (1)/(x + n - 1)	Exp[x]*ExpIntegralE[n, x] <= Divide[1,x + n - 1]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E22	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\frac{E_{n}\left(x\right)}{E_{n-1}\left(x\right)}>0}}$ \deriv{}{x}\frac{\genexpintE{n}@{x}}{\genexpintE{n-1}@{x}} > 0		diff((Ei(n, x))/(Ei(n - 1, x)), x) > 0	D[Divide[ExpIntegralE[n, x],ExpIntegralE[n - 1, x]], x] > 0	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
8.19.E23	${\displaystyle{\displaystyle\int_{z}^{\infty}E_{p-1}\left(t\right)\mathrm{d}t=E_{p}\left(z\right)}}$ \int_{z}^{\infty}\genexpintE{p-1}@{t}\diff{t} = \genexpintE{p}@{z}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}$	int(Ei(p - 1, t), t = z..infinity) = Ei(p, z)	Integrate[ExpIntegralE[p - 1, t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[p, z]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 70]
8.19.E24	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}E_{n}\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}}{a^{n}}\left(\ln\left(1+a\right)+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right)}}$ \int_{0}^{\infty}e^{-at}\genexpintE{n}@{t}\diff{t} = \frac{(-1)^{n-1}}{a^{n}}\left(\ln@{1+a}+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right)	${\displaystyle{\displaystyle\Re a>-1}}$	int(exp(- a*t)*Ei(n, t), t = 0..infinity) = ((- 1)^(n - 1))/((a)^(n))*(ln(1 + a)+ sum(((- 1)^(k)* (a)^(k))/(k), k = 1..n - 1))	Integrate[Exp[- a*t]*ExpIntegralE[n, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(- 1)^(n - 1),(a)^(n)]*(Log[1 + a]+ Sum[Divide[(- 1)^(k)* (a)^(k),k], {k, 1, n - 1}, GenerateConditions->None])	Successful	Failure	-	Successful [Tested: 18]
8.19.E25	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+a)\right)}}$ \int_{0}^{\infty}e^{-at}t^{b-1}\genexpintE{p}@{t}\diff{t} = \frac{\EulerGamma@{b}(1+a)^{-b}}{p+b-1}\*\hyperF@{1}{b}{p+b}{a/(1+a)}	${\displaystyle{\displaystyle\Re a>-1,\Re\left(p+b\right)>1,\Re b>0}}$	int(exp(- a*t)*(t)^(b - 1)* Ei(p, t), t = 0..infinity) = (GAMMA(b)*(1 + a)^(- b))/(p + b - 1)* hypergeom([1, b], [p + b], a/(1 + a))	Integrate[Exp[- a*t]*(t)^(b - 1)* ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[b]*(1 + a)^(- b),p + b - 1]* Hypergeometric2F1[1, b, p + b, a/(1 + a)]	Failure	Aborted	Skipped - Because timed out	Successful [Tested: 64]
8.19.E26	${\displaystyle{\displaystyle\int_{0}^{\infty}E_{p}\left(t\right)E_{q}\left(t\right)\mathrm{d}t=\frac{L(p)+L(q)}{p+q-1}}}$ \int_{0}^{\infty}\genexpintE{p}@{t}\genexpintE{q}@{t}\diff{t} = \frac{L(p)+L(q)}{p+q-1}	${\displaystyle{\displaystyle p>0,q>0,p+q>1}}$	int(Ei(p, t)*Ei(q, t), t = 0..infinity) = (L(p)+ L(q))/(p + q - 1)	Integrate[ExpIntegralE[p, t]*ExpIntegralE[q, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[L[p]+ L[q],p + q - 1]	Failure	Failure	Skipped - Because timed out	Failed [80 / 80] Result: Complex[-0.8698344324715543, -0.7499999999999999] Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 1.5]} Result: Complex[0.26794919243112303, -0.9999999999999999] Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 0.5]} ... skip entries to safe data
8.19.E27	${\displaystyle{\displaystyle L(p)=\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)\mathrm{d}t}}$ L(p) = \int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t}	${\displaystyle{\displaystyle p>0}}$	L(p) = int(exp(- t)*Ei(p, t), t = 0..infinity)	L[p] == Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [30 / 30] Result: Complex[0.8698344324715546, 0.7499999999999999] Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5]} Result: Complex[-1.1377836249026771, 0.24999999999999997] Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 0.5]} ... skip entries to safe data
8.19.E27	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)\mathrm{d}t=\frac{1}{2p}F\left(1,1;1+p;\tfrac{1}{2}\right)}}$ \int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t} = \frac{1}{2p}\hyperF@{1}{1}{1+p}{\tfrac{1}{2}}	${\displaystyle{\displaystyle p>0}}$	int(exp(- t)*Ei(p, t), t = 0..infinity) = (1)/(2*p)*hypergeom([1, 1], [1 + p], (1)/(2))	Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*p]*Hypergeometric2F1[1, 1, 1 + p, Divide[1,2]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]