Exponential, Logarithmic, Sine, and Cosine Integrals - 6.4 Analytic Continuation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.4.E1 E 1 ⁡ ( z ) = Ein ⁡ ( z ) - Ln ⁡ z - γ exponential-integral 𝑧 complementary-exponential-integral 𝑧 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-% \operatorname{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
6.4.E2 E 1 ⁡ ( z ⁢ e 2 ⁢ m ⁢ π ⁢ i ) = E 1 ⁡ ( z ) - 2 ⁢ m ⁢ π ⁢ i exponential-integral 𝑧 superscript 𝑒 2 𝑚 𝜋 𝑖 exponential-integral 𝑧 2 𝑚 𝜋 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right% )-2m\pi i}}
\expintE@{ze^{2m\pi i}} = \expintE@{z}-2m\pi i		 

Ei(z*exp(2*m*Pi*I)) = Ei(z)- 2*m*Pi*I

ExpIntegralE[1, z*Exp[2*m*Pi*I]] == ExpIntegralE[1, z]- 2*m*Pi*I
Failure Failure
Failed [21 / 21]
Result: -.1e-8+6.283185310*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 1, m = 3}


Result: -.6e-8+12.56637063*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, m = 2, m = 3}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.0, 18.84955592153876]
Test Values: {Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, 18.84955592153876]
Test Values: {Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
6.4.E3 E 1 ⁡ ( z ⁢ e + π ⁢ i ) = Ein ⁡ ( - z ) - ln ⁡ z - γ - π ⁢ i exponential-integral 𝑧 superscript 𝑒 𝜋 𝑖 complementary-exponential-integral 𝑧 𝑧 𝜋 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{+\pi i}\right)=\mathrm{Ein}\left(-% z\right)-\ln z-\gamma-\pi i}}
\expintE@{ze^{+\pi i}} = \expintEin@{-z}-\ln@@{z}-\EulerConstant-\pi i		 | ph ⁡ z | ≤ π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi}} 
Error

ExpIntegralE[1, z*Exp[+ Pi*I]] == ExpIntegralE[1, - z] + Ln[- z] + EulerGamma - Log[z]- EulerGamma - Pi*I
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.0, 3.6651914291880923], 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, 5.235987755982989], 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
6.4.E3 E 1 ⁡ ( z ⁢ e - π ⁢ i ) = Ein ⁡ ( - z ) - ln ⁡ z - γ + π ⁢ i exponential-integral 𝑧 superscript 𝑒 𝜋 𝑖 complementary-exponential-integral 𝑧 𝑧 𝜋 𝑖 {\displaystyle{\displaystyle E_{1}\left(ze^{-\pi i}\right)=\mathrm{Ein}\left(-% z\right)-\ln z-\gamma+\pi i}}
\expintE@{ze^{-\pi i}} = \expintEin@{-z}-\ln@@{z}-\EulerConstant+\pi i		 | ph ⁡ z | ≤ π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi}} 
Error

ExpIntegralE[1, z*Exp[- Pi*I]] == ExpIntegralE[1, - z] + Ln[- z] + EulerGamma - Log[z]- EulerGamma + Pi*I
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.0, -2.6179938779914944], 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, -1.0471975511965976], 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
6.4.E4 Ci ⁡ ( z ⁢ e + π ⁢ i ) = + π ⁢ i + Ci ⁡ ( z ) cosine-integral 𝑧 superscript 𝑒 𝜋 𝑖 𝜋 𝑖 cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(ze^{+\pi i}\right)=+\pi i+\mathrm% {Ci}\left(z\right)}}
\cosint@{ze^{+\pi i}} = +\pi i+\cosint@{z}		 

Ci(z*exp(+ Pi*I)) = + Pi*I + Ci(z)

CosIntegral[z*Exp[+ Pi*I]] == + Pi*I + CosIntegral[z]
Failure Failure
Failed [2 / 7]
Result: 0.-6.283185308*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 0.-6.283185308*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

Failed [2 / 7]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

6.4.E4 Ci ⁡ ( z ⁢ e - π ⁢ i ) = - π ⁢ i + Ci ⁡ ( z ) cosine-integral 𝑧 superscript 𝑒 𝜋 𝑖 𝜋 𝑖 cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(ze^{-\pi i}\right)=-\pi i+\mathrm% {Ci}\left(z\right)}}
\cosint@{ze^{-\pi i}} = -\pi i+\cosint@{z}		 

Ci(z*exp(- Pi*I)) = - Pi*I + Ci(z)

CosIntegral[z*Exp[- Pi*I]] == - Pi*I + CosIntegral[z]
Failure Failure
Failed [5 / 7]
Result: 0.+6.283185308*I
Test Values: {z = 1/2-1/2*I*3^(1/2)}


Result: 0.+6.283185308*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [5 / 7]
Result: Complex[0.0, 6.283185307179585]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
6.4.E5 Chi ⁡ ( z ⁢ e + π ⁢ i ) = + π ⁢ i + Chi ⁡ ( z ) hyperbolic-cosine-integral 𝑧 superscript 𝑒 𝜋 𝑖 𝜋 𝑖 hyperbolic-cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Chi}\left(ze^{+\pi i}\right)=+\pi i+% \mathrm{Chi}\left(z\right)}}
\coshint@{ze^{+\pi i}} = +\pi i+\coshint@{z}		 

Chi(z*exp(+ Pi*I)) = + Pi*I + Chi(z)

CoshIntegral[z*Exp[+ Pi*I]] == + Pi*I + CoshIntegral[z]
Failure Failure
Failed [2 / 7]
Result: 0.-6.283185308*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 0.-6.283185307*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

Failed [2 / 7]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

6.4.E5 Chi ⁡ ( z ⁢ e - π ⁢ i ) = - π ⁢ i + Chi ⁡ ( z ) hyperbolic-cosine-integral 𝑧 superscript 𝑒 𝜋 𝑖 𝜋 𝑖 hyperbolic-cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Chi}\left(ze^{-\pi i}\right)=-\pi i+% \mathrm{Chi}\left(z\right)}}
\coshint@{ze^{-\pi i}} = -\pi i+\coshint@{z}		 

Chi(z*exp(- Pi*I)) = - Pi*I + Chi(z)

CoshIntegral[z*Exp[- Pi*I]] == - Pi*I + CoshIntegral[z]
Failure Failure
Failed [5 / 7]
Result: 0.+6.283185307*I
Test Values: {z = 1/2-1/2*I*3^(1/2)}


Result: 0.+6.283185308*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [5 / 7]
Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
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