Exponential, Logarithmic, Sine, and Cosine Integrals - 6.5 Further Interrelations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.5.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \expintE@{-x+ i0} = -\expintEi@{x}- i\pi}
\expintE@{-x+ i0} = -\expintEi@{x}- i\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

ExpIntegralE[1, - x + I*0] == - ExpIntegralEi[x]- I*Pi
Missing Macro Error Failure - Successful [Tested: 3]
6.5.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \expintE@{-x- i0} = -\expintEi@{x}+ i\pi}
\expintE@{-x- i0} = -\expintEi@{x}+ i\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \expintEi@{x} = -\tfrac{1}{2}(\expintE@{-x+i0}+\expintE@{-x-i0})}
\expintEi@{x} = -\tfrac{1}{2}(\expintE@{-x+i0}+\expintE@{-x-i0})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tfrac{1}{2}(\expintEi@{x}+\expintE@{x}) = \sinhint@{x}}
\tfrac{1}{2}(\expintEi@{x}+\expintE@{x}) = \sinhint@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Divide[1,2]*(ExpIntegralEi[x]+ ExpIntegralE[1, x]) == SinhIntegral[x]
Missing Macro Error Failure - Successful [Tested: 3]
6.5.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sinhint@{x} = -i\sinint@{ix}}
\sinhint@{x} = -i\sinint@{ix}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Shi(x) = - I*Si(I*x)

SinhIntegral[x] == - I*SinIntegral[I*x]
Successful Successful - Successful [Tested: 3]
6.5.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tfrac{1}{2}(\expintEi@{x}-\expintE@{x}) = \coshint@{x}}
\tfrac{1}{2}(\expintEi@{x}-\expintE@{x}) = \coshint@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \coshint@{x} = \cosint@{ix}-\tfrac{1}{2}\pi i}
\coshint@{x} = \cosint@{ix}-\tfrac{1}{2}\pi i		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sinint@{z} = \tfrac{1}{2}i(\expintE@{-iz}-\expintE@{iz})+\tfrac{1}{2}\pi}
\sinint@{z} = \tfrac{1}{2}i(\expintE@{-iz}-\expintE@{iz})+\tfrac{1}{2}\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cosint@{z} = -\tfrac{1}{2}(\expintE@{iz}+\expintE@{-iz})}
\cosint@{z} = -\tfrac{1}{2}(\expintE@{iz}+\expintE@{-iz})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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]]]}

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