6.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
6.5.E1	${\displaystyle{\displaystyle E_{1}\left(-x+i0\right)=-\mathrm{Ei}\left(x\right)-i\pi}}$ \expintE@{-x+ i0} = -\expintEi@{x}- i\pi		Error	ExpIntegralE[1, - x + I*0] == - ExpIntegralEi[x]- I*Pi	Missing Macro Error	Failure	-	Successful [Tested: 3]
6.5.E1	${\displaystyle{\displaystyle E_{1}\left(-x-i0\right)=-\mathrm{Ei}\left(x\right)+i\pi}}$ \expintE@{-x- i0} = -\expintEi@{x}+ i\pi		Error	ExpIntegralE[1, - x - I*0] == - ExpIntegralEi[x]+ I*Pi	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[x, 1.5]} Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
6.5.E2	${\displaystyle{\displaystyle\mathrm{Ei}\left(x\right)=-\tfrac{1}{2}(E_{1}\left(-x+i0\right)+E_{1}\left(-x-i0\right))}}$ \expintEi@{x} = -\tfrac{1}{2}(\expintE@{-x+i0}+\expintE@{-x-i0})		Error	ExpIntegralEi[x] == -Divide[1,2]*(ExpIntegralE[1, - x + I*0]+ ExpIntegralE[1, - x - I*0])	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Complex[0.0, -3.141592653589793] Test Values: {Rule[x, 1.5]} Result: Complex[0.0, -3.141592653589793] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
6.5.E3	${\displaystyle{\displaystyle\tfrac{1}{2}(\mathrm{Ei}\left(x\right)+E_{1}\left(x\right))=\mathrm{Shi}\left(x\right)}}$ \tfrac{1}{2}(\expintEi@{x}+\expintE@{x}) = \sinhint@{x}		Error	Divide[1,2]*(ExpIntegralEi[x]+ ExpIntegralE[1, x]) == SinhIntegral[x]	Missing Macro Error	Failure	-	Successful [Tested: 3]
6.5.E3	${\displaystyle{\displaystyle\mathrm{Shi}\left(x\right)=-i\mathrm{Si}\left(ix\right)}}$ \sinhint@{x} = -i\sinint@{ix}		Shi(x) = - I*Si(I*x)	SinhIntegral[x] == - I*SinIntegral[I*x]	Successful	Successful	-	Successful [Tested: 3]
6.5.E4	${\displaystyle{\displaystyle\tfrac{1}{2}(\mathrm{Ei}\left(x\right)-E_{1}\left(x\right))=\mathrm{Chi}\left(x\right)}}$ \tfrac{1}{2}(\expintEi@{x}-\expintE@{x}) = \coshint@{x}		Error	Divide[1,2]*(ExpIntegralEi[x]- ExpIntegralE[1, x]) == CoshIntegral[x]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 3]
6.5.E4	${\displaystyle{\displaystyle\mathrm{Chi}\left(x\right)=\mathrm{Ci}\left(ix\right)-\tfrac{1}{2}\pi i}}$ \coshint@{x} = \cosint@{ix}-\tfrac{1}{2}\pi i		Chi(x) = Ci(I*x)-(1)/(2)*Pi*I	CoshIntegral[x] == CosIntegral[I*x]-Divide[1,2]*Pi*I	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
6.5.E5	${\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\tfrac{1}{2}i(E_{1}\left(-iz\right)-E_{1}\left(iz\right))+\tfrac{1}{2}\pi}}$ \sinint@{z} = \tfrac{1}{2}i(\expintE@{-iz}-\expintE@{iz})+\tfrac{1}{2}\pi		Si(z) = (1)/(2)*I*(Ei(- I*z)- Ei(I*z))+(1)/(2)*Pi	SinIntegral[z] == Divide[1,2]*I*(ExpIntegralE[1, - I*z]- ExpIntegralE[1, I*z])+Divide[1,2]*Pi	Failure	Failure	Failed [5 / 7] Result: -3.141592653+0.*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -3.141592654-.1e-9*I Test Values: {z = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Failed [2 / 7] Result: Complex[-3.141592653589793, 0.0] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-3.141592653589793, 0.0] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
6.5.E6	${\displaystyle{\displaystyle\mathrm{Ci}\left(z\right)=-\tfrac{1}{2}(E_{1}\left(iz\right)+E_{1}\left(-iz\right))}}$ \cosint@{z} = -\tfrac{1}{2}(\expintE@{iz}+\expintE@{-iz})		Ci(z) = -(1)/(2)*(Ei(I*z)+ Ei(- I*z))	CosIntegral[z] == -Divide[1,2]*(ExpIntegralE[1, I*z]+ ExpIntegralE[1, - I*z])	Failure	Failure	Failed [7 / 7] Result: .8944744988+.632221722*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: 1.393548628+1.498247032*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [2 / 7] Result: Complex[0.0, 3.141592653589793] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, -3.141592653589793] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}