Gamma Function - 6.2 Definitions and Interrelations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.2.E1 E 1 ⁡ ( z ) = ∫ z ∞ e - t t ⁢ d t exponential-integral 𝑧 superscript subscript 𝑧 superscript 𝑒 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle E_{1}\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}% }{t}\mathrm{d}t}}
\expintE@{z} = \int_{z}^{\infty}\frac{e^{-t}}{t}\diff{t}		 z ≠ 0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}} 
Ei(z) = int((exp(- t))/(t), t = z..infinity)

ExpIntegralE[1, z] == Integrate[Divide[Exp[- t],t], {t, z, 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]
6.2.E2 E 1 ⁡ ( z ) = e - z ⁢ ∫ 0 ∞ e - t t + z ⁢ d t exponential-integral 𝑧 superscript 𝑒 𝑧 superscript subscript 0 superscript 𝑒 𝑡 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{% e^{-t}}{t+z}\mathrm{d}t}}
\expintE@{z} = e^{-z}\int_{0}^{\infty}\frac{e^{-t}}{t+z}\diff{t}		 | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}} 
Ei(z) = exp(- z)*int((exp(- t))/(t + z), t = 0..infinity)

ExpIntegralE[1, z] == Exp[- z]*Integrate[Divide[Exp[- t],t + z], {t, 0, 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]
6.2.E3 Ein ⁡ ( z ) = ∫ 0 z 1 - e - t t ⁢ d t complementary-exponential-integral 𝑧 superscript subscript 0 𝑧 1 superscript 𝑒 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{% -t}}{t}\mathrm{d}t}}
\expintEin@{z} = \int_{0}^{z}\frac{1-e^{-t}}{t}\diff{t}		 

Error

ExpIntegralE[1, z] + Ln[z] + EulerGamma == Integrate[Divide[1 - Exp[- t],t], {t, 0, z}, 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
6.2.E4 E 1 ⁡ ( z ) = Ein ⁡ ( z ) - ln ⁡ z - γ exponential-integral 𝑧 complementary-exponential-integral 𝑧 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-% \ln z-\gamma}}
\expintE@{z} = \expintEin@{z}-\ln@@{z}-\EulerConstant		 

Error

ExpIntegralE[1, z] == ExpIntegralE[1, z] + Ln[z] + EulerGamma - Log[z]- EulerGamma
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.0, 0.5235987755982988], Times[-1.0, 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], Times[-1.0, Ln[Complex[-0.4999999999999998, 0.8660254037844387]]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
6.2.E6 Ei ⁡ ( - x ) = - ∫ x ∞ e - t t ⁢ d t exponential-integral-Ei 𝑥 superscript subscript 𝑥 superscript 𝑒 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Ei}\left(-x\right)=-\int_{x}^{\infty}\frac% {e^{-t}}{t}\mathrm{d}t}}
\expintEi@{-x} = -\int_{x}^{\infty}\frac{e^{-t}}{t}\diff{t}		 

Error

ExpIntegralEi[- x] == - Integrate[Divide[Exp[- t],t], {t, x, Infinity}, GenerateConditions->None]
Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 3]
6.2.E6 - ∫ x ∞ e - t t ⁢ d t = - E 1 ⁡ ( x ) superscript subscript 𝑥 superscript 𝑒 𝑡 𝑡 𝑡 exponential-integral 𝑥 {\displaystyle{\displaystyle-\int_{x}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t=-E_{% 1}\left(x\right)}}
-\int_{x}^{\infty}\frac{e^{-t}}{t}\diff{t} = -\expintE@{x}		 

- int((exp(- t))/(t), t = x..infinity) = - Ei(x)

- Integrate[Divide[Exp[- t],t], {t, x, Infinity}, GenerateConditions->None] == - ExpIntegralE[1, x]
Failure Failure
Failed [3 / 3]
Result: 3.201265867
Test Values: {x = 1.5}


Result: -.1055536899
Test Values: {x = .5}

... skip entries to safe data
 Successful [Tested: 3]
6.2.E7 Ei ⁡ ( + x ) = - Ein ⁡ ( - x ) + ln ⁡ x + γ exponential-integral-Ei 𝑥 complementary-exponential-integral 𝑥 𝑥 {\displaystyle{\displaystyle\mathrm{Ei}\left(+x\right)=-\mathrm{Ein}\left(-x% \right)+\ln x+\gamma}}
\expintEi@{+ x} = -\expintEin@{- x}+\ln@@{x}+\EulerConstant		 

Error

ExpIntegralEi[+ x] == - ExpIntegralE[1, - x] + Ln[- x] + EulerGamma + Log[x]+ EulerGamma
Missing Macro Error Failure -
Failed [3 / 3]
Result: Plus[Complex[-1.5598964379112301, -3.141592653589793], Times[-1.0, Ln[-1.5]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[Complex[-0.46128414924312044, -3.141592653589793], Times[-1.0, Ln[-0.5]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
6.2.E7 Ei ⁡ ( - x ) = - Ein ⁡ ( + x ) + ln ⁡ x + γ exponential-integral-Ei 𝑥 complementary-exponential-integral 𝑥 𝑥 {\displaystyle{\displaystyle\mathrm{Ei}\left(-x\right)=-\mathrm{Ein}\left(+x% \right)+\ln x+\gamma}}
\expintEi@{- x} = -\expintEin@{+ x}+\ln@@{x}+\EulerConstant		 

Error

ExpIntegralEi[- x] == - ExpIntegralE[1, + x] + Ln[+ x] + EulerGamma + Log[x]+ EulerGamma
Missing Macro Error Failure -
Failed [3 / 3]
Result: Plus[-1.5598964379112301, Times[-1.0, Ln[1.5]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.46128414924312044, Times[-1.0, Ln[0.5]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
6.2.E9 Si ⁡ ( z ) = ∫ 0 z sin ⁡ t t ⁢ d t sine-integral 𝑧 superscript subscript 0 𝑧 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Si}\left(z\right)=\int_{0}^{z}\frac{\sin t% }{t}\mathrm{d}t}}
\sinint@{z} = \int_{0}^{z}\frac{\sin@@{t}}{t}\diff{t}		 

Si(z) = int((sin(t))/(t), t = 0..z)

SinIntegral[z] == Integrate[Divide[Sin[t],t], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
6.2.E10 si ⁡ ( z ) = - ∫ z ∞ sin ⁡ t t ⁢ d t shifted-sine-integral 𝑧 superscript subscript 𝑧 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{si}\left(z\right)=-\int_{z}^{\infty}\frac{% \sin t}{t}\mathrm{d}t}}
\shiftsinint@{z} = -\int_{z}^{\infty}\frac{\sin@@{t}}{t}\diff{t}		 

Ssi(z) = - int((sin(t))/(t), t = z..infinity)

SinIntegral[z] - Pi/2 == - Integrate[Divide[Sin[t],t], {t, z, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
6.2.E10 - ∫ z ∞ sin ⁡ t t ⁢ d t = Si ⁡ ( z ) - 1 2 ⁢ π superscript subscript 𝑧 𝑡 𝑡 𝑡 sine-integral 𝑧 1 2 𝜋 {\displaystyle{\displaystyle-\int_{z}^{\infty}\frac{\sin t}{t}\mathrm{d}t=% \mathrm{Si}\left(z\right)-\tfrac{1}{2}\pi}}
-\int_{z}^{\infty}\frac{\sin@@{t}}{t}\diff{t} = \sinint@{z}-\tfrac{1}{2}\pi		 

- int((sin(t))/(t), t = z..infinity) = Si(z)-(1)/(2)*Pi

- Integrate[Divide[Sin[t],t], {t, z, Infinity}, GenerateConditions->None] == SinIntegral[z]-Divide[1,2]*Pi
Successful Successful - Successful [Tested: 7]
6.2.E11 Ci ⁡ ( z ) = - ∫ z ∞ cos ⁡ t t ⁢ d t cosine-integral 𝑧 superscript subscript 𝑧 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Ci}(z)=-\int_{z}^{\infty}\frac{\cos t}{t}% \mathrm{d}t}}
\cosint(z) = -\int_{z}^{\infty}\frac{\cos@@{t}}{t}\diff{t}		 

Ci((z) ) = - int((cos(t))/(t), t = z..infinity)

CosIntegral[(z) ] == - Integrate[Divide[Cos[t],t], {t, z, Infinity}, GenerateConditions->None]
Translation Error Translation Error - -
6.2#Ex1 lim x → ∞ ⁡ Si ⁡ ( x ) = 1 2 ⁢ π subscript → 𝑥 sine-integral 𝑥 1 2 𝜋 {\displaystyle{\displaystyle\lim_{x\to\infty}\mathrm{Si}\left(x\right)=\tfrac{% 1}{2}\pi}}
\lim_{x\to\infty}\sinint@{x} = \tfrac{1}{2}\pi		 

limit(Si(x), x = infinity) = (1)/(2)*Pi

Limit[SinIntegral[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]*Pi
Successful Successful - Successful [Tested: 1]
6.2#Ex2 lim x → ∞ ⁡ Ci ⁡ ( x ) = 0 subscript → 𝑥 cosine-integral 𝑥 0 {\displaystyle{\displaystyle\lim_{x\to\infty}\mathrm{Ci}\left(x\right)=0}}
\lim_{x\to\infty}\cosint@{x} = 0		 

limit(Ci(x), x = infinity) = 0

Limit[CosIntegral[x], x -> Infinity, GenerateConditions->None] == 0
Successful Successful - Successful [Tested: 1]
6.2.E15 Shi ⁡ ( z ) = ∫ 0 z sinh ⁡ t t ⁢ d t hyperbolic-sine-integral 𝑧 superscript subscript 0 𝑧 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Shi}\left(z\right)=\int_{0}^{z}\frac{\sinh t% }{t}\mathrm{d}t}}
\sinhint@{z} = \int_{0}^{z}\frac{\sinh@@{t}}{t}\diff{t}		 

Shi(z) = int((sinh(t))/(t), t = 0..z)

SinhIntegral[z] == Integrate[Divide[Sinh[t],t], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
6.2.E16 Chi ⁡ ( z ) = γ + ln ⁡ z + ∫ 0 z cosh ⁡ t - 1 t ⁢ d t hyperbolic-cosine-integral 𝑧 𝑧 superscript subscript 0 𝑧 𝑡 1 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Chi}\left(z\right)=\gamma+\ln z+\int_{0}^{% z}\frac{\cosh t-1}{t}\mathrm{d}t}}
\coshint@{z} = \EulerConstant+\ln@@{z}+\int_{0}^{z}\frac{\cosh@@{t}-1}{t}\diff{t}		 

Chi(z) = gamma + ln(z)+ int((cosh(t)- 1)/(t), t = 0..z)

CoshIntegral[z] == EulerGamma + Log[z]+ Integrate[Divide[Cosh[t]- 1,t], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
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