Elementary Functions - 4.37 Inverse Hyperbolic Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.37.E1 Arcsinh ⁑ z = ∫ 0 z d t ( 1 + t 2 ) 1 / 2 multivalued-hyperbolic-inverse-sine 𝑧 superscript subscript 0 𝑧 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}z=\int_{0}^{z}\frac{\mathrm{% d}t}{(1+t^{2})^{1/2}}}}
\Asinh@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1+t^{2})^{1/2}}		 

Error

ArcSinh[z] == Integrate[Divide[1,(1 + (t)^(2))^(1/2)], {t, 0, z}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
4.37.E2 Arccosh ⁑ z = ∫ 1 z d t ( t 2 - 1 ) 1 / 2 multivalued-hyperbolic-inverse-cosine 𝑧 superscript subscript 1 𝑧 𝑑 superscript superscript 𝑑 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}z=\int_{1}^{z}\frac{\mathrm{% d}t}{(t^{2}-1)^{1/2}}}}
\Acosh@@{z} = \int_{1}^{z}\frac{\diff{t}}{(t^{2}-1)^{1/2}}		 

Error

ArcCosh[z] == Integrate[Divide[1,((t)^(2)- 1)^(1/2)], {t, 1, z}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
4.37.E3 Arctanh ⁑ z = ∫ 0 z d t 1 - t 2 multivalued-hyperbolic-inverse-tangent 𝑧 superscript subscript 0 𝑧 𝑑 1 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{Arctanh}z=\int_{0}^{z}\frac{\mathrm{% d}t}{1-t^{2}}}}
\Atanh@@{z} = \int_{0}^{z}\frac{\diff{t}}{1-t^{2}}		 

Error

ArcTanh[z] == Integrate[Divide[1,1 - (t)^(2)], {t, 0, z}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 1]
4.37.E4 Arccsch ⁑ z = Arcsinh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-cosecant 𝑧 multivalued-hyperbolic-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccsch}z=\operatorname{Arcsinh}% \left(1/z\right)}}
\Acsch@@{z} = \Asinh@{1/z}		 

Error

ArcCsch[z] == ArcSinh[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.37.E5 Arcsech ⁑ z = Arccosh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-secant 𝑧 multivalued-hyperbolic-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arcsech}z=\operatorname{Arccosh}% \left(1/z\right)}}
\Asech@@{z} = \Acosh@{1/z}		 

Error

ArcSech[z] == ArcCosh[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.37.E6 Arccoth ⁑ z = Arctanh ⁑ ( 1 / z ) multivalued-hyperbolic-inverse-cotangent 𝑧 multivalued-hyperbolic-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccoth}z=\operatorname{Arctanh}% \left(1/z\right)}}
\Acoth@@{z} = \Atanh@{1/z}		 

Error

ArcCoth[z] == ArcTanh[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.37.E7 arccsch ⁑ z = arcsinh ⁑ ( 1 / z ) hyperbolic-inverse-cosecant 𝑧 hyperbolic-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccsch}z=\operatorname{arcsinh}% \left(1/z\right)}}
\acsch@@{z} = \asinh@{1/z}		 

arccsch(z) = arcsinh(1/z)

ArcCsch[z] == ArcSinh[1/z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.37.E8 arcsech ⁑ z = arccosh ⁑ ( 1 / z ) hyperbolic-inverse-secant 𝑧 hyperbolic-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}z=\operatorname{arccosh}% \left(1/z\right)}}
\asech@@{z} = \acosh@{1/z}		 

arcsech(z) = arccosh(1/z)

ArcSech[z] == ArcCosh[1/z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.37.E9 arccoth ⁑ z = arctanh ⁑ ( 1 / z ) hyperbolic-inverse-cotangent 𝑧 hyperbolic-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccoth}z=\operatorname{arctanh}% \left(1/z\right)}}
\acoth@@{z} = \atanh@{1/z}		 

arccoth(z) = arctanh(1/z)

ArcCoth[z] == ArcTanh[1/z]
Failure Successful Successful [Tested: 7] Successful [Tested: 1]
4.37.E10 arcsinh ⁑ ( - z ) = - arcsinh ⁑ z hyperbolic-inverse-sine 𝑧 hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(-z\right)=-% \operatorname{arcsinh}z}}
\asinh@{-z} = -\asinh@@{z}		 

arcsinh(- z) = - arcsinh(z)

ArcSinh[- z] == - ArcSinh[z]
Successful Successful - Successful [Tested: 7]
4.37.E11 arccosh ⁑ ( - z ) = + Ο€ ⁒ i + arccosh ⁑ z hyperbolic-inverse-cosine 𝑧 πœ‹ 𝑖 hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=+\pi i+% \operatorname{arccosh}z}}
\acosh@{-z} = +\pi i+\acosh@@{z}		 

arccosh(- z) = + Pi*I + arccosh(z)

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


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


Result: -2.094395103*I
Test Values: {z = .5, Im(z) = 1/2}

Failed [1 / 1]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[z, Complex[0, Rational[1, 2]]]}

4.37.E11 arccosh ⁑ ( - z ) = - Ο€ ⁒ i + arccosh ⁑ z hyperbolic-inverse-cosine 𝑧 πœ‹ 𝑖 hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=-\pi i+% \operatorname{arccosh}z}}
\acosh@{-z} = -\pi i+\acosh@@{z}		 

arccosh(- z) = - Pi*I + arccosh(z)

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


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


Result: 0.+6.283185308*I
Test Values: {z = 1.5, Im(z) = 1/2}


Result: 4.188790205*I
Test Values: {z = .5, Im(z) = 1/2}

... skip entries to safe data
 Successful [Tested: 1]
4.37.E12 arctanh ⁑ ( - z ) = - arctanh ⁑ z hyperbolic-inverse-tangent 𝑧 hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\left(-z\right)=-% \operatorname{arctanh}z}}
\atanh@{-z} = -\atanh@@{z}		 

arctanh(- z) = - arctanh(z)

ArcTanh[- z] == - ArcTanh[z]
Successful Successful - Successful [Tested: 1]
4.37.E13 arccsch ⁑ ( - z ) = - arccsch ⁑ z hyperbolic-inverse-cosecant 𝑧 hyperbolic-inverse-cosecant 𝑧 {\displaystyle{\displaystyle\operatorname{arccsch}\left(-z\right)=-% \operatorname{arccsch}z}}
\acsch@{-z} = -\acsch@@{z}		 

arccsch(- z) = - arccsch(z)

ArcCsch[- z] == - ArcCsch[z]
Successful Successful - Successful [Tested: 7]
4.37.E14 arcsech ⁑ ( - z ) = - Ο€ ⁒ i + arcsech ⁑ z hyperbolic-inverse-secant 𝑧 πœ‹ 𝑖 hyperbolic-inverse-secant 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=-\pi i+% \operatorname{arcsech}z}}
\asech@{-z} = -\pi i+\asech@@{z}		 

arcsech(- z) = - Pi*I + arcsech(z)

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


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


Result: 4.601047966*I
Test Values: {z = 1.5, Im(z) = 1/2}


Result: 0.+6.283185308*I
Test Values: {z = .5, Im(z) = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[z, Complex[0, Rational[1, 2]]]}

4.37.E14 arcsech ⁑ ( - z ) = + Ο€ ⁒ i + arcsech ⁑ z hyperbolic-inverse-secant 𝑧 πœ‹ 𝑖 hyperbolic-inverse-secant 𝑧 {\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=+\pi i+% \operatorname{arcsech}z}}
\asech@{-z} = +\pi i+\asech@@{z}		 

arcsech(- z) = + Pi*I + arcsech(z)

ArcSech[- z] == + Pi*I + ArcSech[z]
Failure Failure
Failed [4 / 7]
Result: 0.-6.283185307*I
Test Values: {z = 1/2-1/2*I*3^(1/2), Im(z) = 1/2}


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


Result: -1.682137342*I
Test Values: {z = 1.5, Im(z) = 1/2}


Result: -2.094395103*I
Test Values: {z = 2, Im(z) = 1/2}

Successful [Tested: 1]
4.37.E15 arccoth ⁑ ( - z ) = - arccoth ⁑ z hyperbolic-inverse-cotangent 𝑧 hyperbolic-inverse-cotangent 𝑧 {\displaystyle{\displaystyle\operatorname{arccoth}\left(-z\right)=-% \operatorname{arccoth}z}}
\acoth@{-z} = -\acoth@@{z}		 

arccoth(- z) = - arccoth(z)

ArcCoth[- z] == - ArcCoth[z]
Failure Successful
Failed [1 / 7]
Result: 0.-3.141592654*I
Test Values: {z = .5, z = 1/2}

Successful [Tested: 1]
4.37.E16 arcsinh ⁑ z = ln ⁑ ( ( z 2 + 1 ) 1 / 2 + z ) hyperbolic-inverse-sine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z% \right)}}
\asinh@@{z} = \ln@{(z^{2}+1)^{1/2}+z}		 

arcsinh(z) = ln(((z)^(2)+ 1)^(1/2)+ z)

ArcSinh[z] == Log[((z)^(2)+ 1)^(1/2)+ z]
Failure Successful Successful [Tested: 7] Successful [Tested: 1]
4.37.E17 arcsinh ⁑ ( i ⁒ y ) = 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 - 1 ) 1 / 2 + y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}% \pi i+\ln\left((y^{2}-1)^{1/2}+y\right)}}
\asinh@{iy} = \tfrac{1}{2}\pi i+\ln@{(y^{2}-1)^{1/2}+y}		 

arcsinh(I*y) = (1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/2)+ y)

ArcSinh[I*y] == Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/2)+ y]
Failure Successful
Failed [4 / 6]
Result: .7e-9-6.283185308*I
Test Values: {y = -1.5, y = 3/2}


Result: -.1347500000e-10-4.188790205*I
Test Values: {y = -.5, y = 3/2}


Result: -.1347500000e-10-2.094395102*I
Test Values: {y = .5, y = 3/2}


Result: .2e-8-6.283185308*I
Test Values: {y = -2, y = 3/2}

Successful [Tested: 1]
4.37.E17 arcsinh ⁑ ( i ⁒ y ) = 1 2 ⁒ Ο€ ⁒ i - ln ⁑ ( ( y 2 - 1 ) 1 / 2 + y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}% \pi i-\ln\left((y^{2}-1)^{1/2}+y\right)}}
\asinh@{iy} = \tfrac{1}{2}\pi i-\ln@{(y^{2}-1)^{1/2}+y}		 

arcsinh(I*y) = (1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/2)+ y)

ArcSinh[I*y] == Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/2)+ y]
Failure Failure
Failed [4 / 6]
Result: -1.924847301+0.*I
Test Values: {y = -1.5, y = 3/2}


Result: 1.924847300+0.*I
Test Values: {y = 1.5, y = 3/2}


Result: -2.633915796+0.*I
Test Values: {y = -2, y = 3/2}


Result: 2.633915794+0.*I
Test Values: {y = 2, y = 3/2}

Failed [1 / 1]
Result: Complex[1.9248473002384139, 0.0]
Test Values: {Rule[y, Rational[3, 2]]}

4.37.E18 arcsinh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 - 1 ) 1 / 2 - y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2% }\pi i+\ln\left((y^{2}-1)^{1/2}-y\right)}}
\asinh@{iy} = -\tfrac{1}{2}\pi i+\ln@{(y^{2}-1)^{1/2}-y}		 

arcsinh(I*y) = -(1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/2)- y)

ArcSinh[I*y] == -Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/2)- y]
Failure Failure
Failed [4 / 6]
Result: -1.924847300+0.*I
Test Values: {y = -1.5, y = -2}


Result: 1.924847301+0.*I
Test Values: {y = 1.5, y = -2}


Result: -2.633915794+0.*I
Test Values: {y = -2, y = -2}


Result: 2.633915796+0.*I
Test Values: {y = 2, y = -2}

Failed [1 / 1]
Result: Complex[-2.633915793849633, 0.0]
Test Values: {Rule[y, -2]}

4.37.E18 arcsinh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i - ln ⁑ ( ( y 2 - 1 ) 1 / 2 - y ) hyperbolic-inverse-sine 𝑖 𝑦 1 2 πœ‹ 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2% }\pi i-\ln\left((y^{2}-1)^{1/2}-y\right)}}
\asinh@{iy} = -\tfrac{1}{2}\pi i-\ln@{(y^{2}-1)^{1/2}-y}		 

arcsinh(I*y) = -(1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/2)- y)

ArcSinh[I*y] == -Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/2)- y]
Failure Failure
Failed [4 / 6]
Result: -.7e-9+6.283185308*I
Test Values: {y = 1.5, y = -2}


Result: .1347500000e-10+2.094395102*I
Test Values: {y = -.5, y = -2}


Result: .1347500000e-10+4.188790205*I
Test Values: {y = .5, y = -2}

Result: -.2e-8+6.283185308*I
Test Values: {y = 2, y = -2}

 Successful [Tested: 1]
4.37.E19 arccosh ⁑ z = ln ⁑ ( + ( z 2 - 1 ) 1 / 2 + z ) hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(+(z^{2}-1)^{1/2}+% z\right)}}
\acosh@@{z} = \ln@{+(z^{2}-1)^{1/2}+z}		 

arccosh(z) = ln(+((z)^(2)- 1)^(1/2)+ z)
 
ArcCosh[z] == Log[+((z)^(2)- 1)^(1/2)+ z]
 Failure Failure
Failed [2 / 7]
Result: 1.662885893+3.891061519*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 3/2}

Result: 1.316957897-4.712388980*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 3/2}

 Successful [Tested: 1]
4.37.E19 arccosh ⁑ z = ln ⁑ ( - ( z 2 - 1 ) 1 / 2 + z ) hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(-(z^{2}-1)^{1/2}+% z\right)}}
\acosh@@{z} = \ln@{-(z^{2}-1)^{1/2}+z}		 

arccosh(z) = ln(-((z)^(2)- 1)^(1/2)+ z)
 
ArcCosh[z] == Log[-((z)^(2)- 1)^(1/2)+ z]
 Failure Failure
Failed [5 / 7]
Result: 1.316957897+1.570796326*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 3/2}

Result: 1.662885893-2.392123788*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 3/2}

Result: 1.924847301
Test Values: {z = 1.5, z = 3/2}

Result: -.1347500000e-10+2.094395102*I
Test Values: {z = .5, z = 3/2}

... skip entries to safe data
Failed [1 / 1]
Result: 1.9248473002384139
Test Values: {Rule[z, Rational[3, 2]]}

4.37.E20 arccosh ⁑ ( i ⁒ y ) = + 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 + 1 ) 1 / 2 + y ) hyperbolic-inverse-cosine imaginary-unit 𝑦 1 2 πœ‹ imaginary-unit superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=+% \tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}+y\right)}}
\acosh@{\iunit y} = +\tfrac{1}{2}\pi\iunit+\ln@{(y^{2}+1)^{1/2}+ y}		 

arccosh(I*y) = +(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/2)+ y)
 
ArcCosh[I*y] == +Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/2)+ y]
 Failure Failure
Failed [3 / 6]
Result: 2.389526433-3.141592654*I
Test Values: {y = -1.5, y = 1/2}

Result: .9624236498-3.141592654*I
Test Values: {y = -.5, y = 1/2}

Result: 2.887270952-3.141592654*I
Test Values: {y = -2, y = 1/2}

 Successful [Tested: 1]
4.37.E20 arccosh ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ ⁒ i + ln ⁑ ( ( y 2 + 1 ) 1 / 2 - y ) hyperbolic-inverse-cosine imaginary-unit 𝑦 1 2 πœ‹ imaginary-unit superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=-% \tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}-y\right)}}
\acosh@{\iunit y} = -\tfrac{1}{2}\pi\iunit+\ln@{(y^{2}+1)^{1/2}- y}		 

arccosh(I*y) = -(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/2)- y)
 
ArcCosh[I*y] == -Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/2)- y]
 Failure Failure
Failed [3 / 6]
Result: 2.389526433+3.141592654*I
Test Values: {y = 1.5, y = 1/2}

Result: .9624236498+3.141592654*I
Test Values: {y = .5, y = 1/2}

Result: 2.887270952+3.141592654*I
Test Values: {y = 2, y = 1/2}
Failed [1 / 1]
Result: Complex[0.9624236501192068, 3.141592653589793]
Test Values: {Rule[y, Rational[1, 2]]}

4.37.E21 arccosh ⁑ z = 2 ⁒ ln ⁑ ( ( z + 1 2 ) 1 / 2 + ( z - 1 2 ) 1 / 2 ) hyperbolic-inverse-cosine 𝑧 2 superscript 𝑧 1 2 1 2 superscript 𝑧 1 2 1 2 {\displaystyle{\displaystyle\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}% {2}\right)^{1/2}+\left(\frac{z-1}{2}\right)^{1/2}\right)}}
\acosh@@{z} = 2\ln@{\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac{z-1}{2}\right)^{1/2}}		 

arccosh(z) = 2*ln(((z + 1)/(2))^(1/2)+((z - 1)/(2))^(1/2))
 
ArcCosh[z] == 2*Log[(Divide[z + 1,2])^(1/2)+(Divide[z - 1,2])^(1/2)]
 Failure Failure Successful [Tested: 7] Successful [Tested: 1]
4.37.E22 arccosh ⁑ x = + ln ⁑ ( i ⁒ ( 1 - x 2 ) 1 / 2 + x ) hyperbolic-inverse-cosine π‘₯ 𝑖 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=+\ln\left(i(1-x^{2})^{1/2}% +x\right)}}
\acosh@@{x} = +\ln@{i(1-x^{2})^{1/2}+x}		 

arccosh(x) = + ln(I*(1 - (x)^(2))^(1/2)+ x)
 
ArcCosh[x] == + Log[I*(1 - (x)^(2))^(1/2)+ x]
 Failure Failure
Failed [2 / 3]
Result: 1.924847301
Test Values: {x = 1.5, x = 1/2}

Result: 2.633915796
Test Values: {x = 2, x = 1/2}

 Successful [Tested: 1]
4.37.E22 arccosh ⁑ x = - ln ⁑ ( i ⁒ ( 1 - x 2 ) 1 / 2 + x ) hyperbolic-inverse-cosine π‘₯ 𝑖 superscript 1 superscript π‘₯ 2 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=-\ln\left(i(1-x^{2})^{1/2}% +x\right)}}
\acosh@@{x} = -\ln@{i(1-x^{2})^{1/2}+x}		 

arccosh(x) = - ln(I*(1 - (x)^(2))^(1/2)+ x)
 
ArcCosh[x] == - Log[I*(1 - (x)^(2))^(1/2)+ x]
 Failure Failure
Failed [1 / 3]
Result: .1347500000e-10+2.094395102*I
Test Values: {x = .5, x = 1/2}
Failed [1 / 1]
Result: Complex[0.0, 2.0943951023931953]
Test Values: {Rule[x, Rational[1, 2]]}

4.37.E23 arccosh ⁑ x = + Ο€ ⁒ i + ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) hyperbolic-inverse-cosine π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=+\pi i+\ln\left((x^{2}-1)^% {1/2}-x\right)}}
\acosh@@{x} = +\pi i+\ln@{(x^{2}-1)^{1/2}-x}		 

arccosh(x) = + Pi*I + ln(((x)^(2)- 1)^(1/2)- x)
 
ArcCosh[x] == + Pi*I + Log[((x)^(2)- 1)^(1/2)- x]
 Failure Failure
Failed [3 / 3]
Result: 1.924847301-6.283185308*I
Test Values: {x = 1.5, x = -2}

Result: -.1347500000e-10-4.188790205*I
Test Values: {x = .5, x = -2}

Result: 2.633915796-6.283185308*I
Test Values: {x = 2, x = -2}

 Successful [Tested: 1]
4.37.E23 arccosh ⁑ x = - Ο€ ⁒ i + ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) hyperbolic-inverse-cosine π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}x=-\pi i+\ln\left((x^{2}-1)^% {1/2}-x\right)}}
\acosh@@{x} = -\pi i+\ln@{(x^{2}-1)^{1/2}-x}		 

arccosh(x) = - Pi*I + ln(((x)^(2)- 1)^(1/2)- x)
 
ArcCosh[x] == - Pi*I + Log[((x)^(2)- 1)^(1/2)- x]
 Failure Failure
Failed [3 / 3]
Result: 1.924847301+0.*I
Test Values: {x = 1.5, x = -2}

Result: -.1347500000e-10+2.094395103*I
Test Values: {x = .5, x = -2}

Result: 2.633915796+0.*I
Test Values: {x = 2, x = -2}
Failed [1 / 1]
Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[x, -2]}

4.37.E24 arctanh ⁑ z = 1 2 ⁒ ln ⁑ ( 1 + z 1 - z ) hyperbolic-inverse-tangent 𝑧 1 2 1 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac% {1+z}{1-z}\right)}}
\atanh@@{z} = \tfrac{1}{2}\ln@{\frac{1+z}{1-z}}		 

arctanh(z) = (1)/(2)*ln((1 + z)/(1 - z))
 
ArcTanh[z] == Divide[1,2]*Log[Divide[1 + z,1 - z]]
 Failure Failure
Failed [2 / 7]
Result: .2e-9-3.141592654*I
Test Values: {z = 1.5, z = 1/2}

Result: -.2e-9-3.141592654*I
Test Values: {z = 2, z = 1/2}

 Successful [Tested: 1]
4.37.E25 arctanh ⁑ x = + 1 2 ⁒ Ο€ ⁒ i + 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) hyperbolic-inverse-tangent π‘₯ 1 2 πœ‹ 𝑖 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\operatorname{arctanh}x=+\tfrac{1}{2}\pi i+\tfrac{% 1}{2}\ln\left(\frac{x+1}{x-1}\right)}}
\atanh@@{x} = +\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln@{\frac{x+1}{x-1}}		 

arctanh(x) = +(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1))
 
ArcTanh[x] == +Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]]
 Failure Failure
Failed [3 / 3]
Result: .2e-9-3.141592654*I
Test Values: {x = 1.5, x = -3/2}

Result: -.2e-9-3.141592654*I
Test Values: {x = .5, x = -3/2}

Result: -.2e-9-3.141592654*I
Test Values: {x = 2, x = -3/2}

 Successful [Tested: 1]
4.37.E25 arctanh ⁑ x = - 1 2 ⁒ Ο€ ⁒ i + 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) hyperbolic-inverse-tangent π‘₯ 1 2 πœ‹ 𝑖 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle\operatorname{arctanh}x=-\tfrac{1}{2}\pi i+\tfrac{% 1}{2}\ln\left(\frac{x+1}{x-1}\right)}}
\atanh@@{x} = -\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln@{\frac{x+1}{x-1}}		 

arctanh(x) = -(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1))
 
ArcTanh[x] == -Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]]
 Failure Failure Successful [Tested: 3]
Failed [1 / 1]
Result: Complex[-1.1102230246251565*^-16, 3.141592653589793]
Test Values: {Rule[x, Rational[-3, 2]]}

4.37.E26 z = sinh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\sinh w}}
z = \sinh@@{w}		 

z = sinh(w)
 
z == Sinh[w]
 Failure Failure
Failed [70 / 70]
Result: .73886869e-2-.1707313589*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -1.358636717+.1952940451*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

Result: -.3586367171-1.536756763*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

Result: -1.724662121-1.170731359*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.007388686967293889, -0.17073135880721174]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-1.3586367168171445, 0.19529404497722702]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.37.E27 z = cosh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\cosh w}}
z = \cosh@@{w}		 

z = cosh(w)
 
z == Cosh[w]
 Failure Failure
Failed [70 / 70]
Result: -.3617401130+.309246236e-1*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -1.727765517+.3969500276*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

Result: -.7277655170-1.335100780*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

Result: -2.093790921-.9690753764*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[-0.3617401130796717, 0.030924623731496126]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-1.7277655168641102, 0.3969500275159349]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.37.E28 z = tanh ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\tanh w}}
z = \tanh@@{w}		 

z = tanh(w)
 
z == Tanh[w]
 Failure Failure
Failed [70 / 70]
Result: .736226475e-1+.2564398629*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

Result: -1.292402756+.6224652669*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

Result: -.2924027565-1.109585541*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

Result: -1.658428160-.7435601371*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.07362264736640245, 0.25643986284286624]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-1.292402756418036, 0.622465266627305]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.37.E29 w = Arcsinh ⁑ z 𝑀 multivalued-hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arcsinh}z}}
w = \Asinh@@{z}		 

Error
 
w == ArcSinh[z]
 Missing Macro Error Failure -
Failed [70 / 70]
Result: Complex[0.03458245825512818, 0.12526556729125993]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.524504352246847, -0.28539816339744856]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.37.E29 Arcsinh ⁑ z = ( - 1 ) k ⁒ arcsinh ⁑ z + k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-sine 𝑧 superscript 1 π‘˜ hyperbolic-inverse-sine 𝑧 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arcsinh}z=(-1)^{k}\operatorname{% arcsinh}z+k\pi i}}
\Asinh@@{z} = (-1)^{k}\asinh@@{z}+k\pi i		 

Error
 
ArcSinh[z] == (- 1)^(k)* ArcSinh[z]+ k*Pi*I
 Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[1.662885891058621, -2.392123788172313]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

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

... skip entries to safe data
4.37.E30 w = Arccosh ⁑ z 𝑀 multivalued-hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arccosh}z}}
w = \Acosh@@{z}		 

Error
 
w == ArcCosh[z]
 Missing Macro Error Failure -
Failed [70 / 70]
Result: Complex[0.20754645532203042, -0.28539816339744833]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.03458245825512796, -1.4455307595036366]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.37.E30 Arccosh ⁑ z = + arccosh ⁑ z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-cosine 𝑧 hyperbolic-inverse-cosine 𝑧 2 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arccosh}z=+\operatorname{arccosh}z+2% k\pi i}}
\Acosh@@{z} = +\acosh@@{z}+2k\pi i		 

Error
 
ArcCosh[z] == + ArcCosh[z]+ 2*k*Pi*I
 Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.0, -12.566370614359172]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.37.E30 Arccosh ⁑ z = - arccosh ⁑ z + 2 ⁒ k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-cosine 𝑧 hyperbolic-inverse-cosine 𝑧 2 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arccosh}z=-\operatorname{arccosh}z+2% k\pi i}}
\Acosh@@{z} = -\acosh@@{z}+2k\pi i		 

Error
 
ArcCosh[z] == - ArcCosh[z]+ 2*k*Pi*I
 Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[1.3169578969248166, -4.71238898038469]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.3169578969248166, -10.995574287564276]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.37.E31 w = Arctanh ⁑ z 𝑀 multivalued-hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arctanh}z}}
w = \Atanh@@{z}		 

Error
 
w == ArcTanh[z]
 Missing Macro Error Failure -
Failed [10 / 10]
Result: Complex[0.3167192594503839, 0.49999999999999994]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Rational[1, 2]]}

Result: Complex[-1.0493061443340546, 0.8660254037844387]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Rational[1, 2]]}

... skip entries to safe data
4.37.E31 Arctanh ⁑ z = arctanh ⁑ z + k ⁒ Ο€ ⁒ i multivalued-hyperbolic-inverse-tangent 𝑧 hyperbolic-inverse-tangent 𝑧 π‘˜ πœ‹ 𝑖 {\displaystyle{\displaystyle\operatorname{Arctanh}z=\operatorname{arctanh}z+k% \pi i}}
\Atanh@@{z} = \atanh@@{z}+k\pi i		 

Error
 
ArcTanh[z] == ArcTanh[z]+ k*Pi*I
 Missing Macro Error Failure -
Failed [3 / 3]
Result: Complex[0.0, -3.141592653589793]
Test Values: {Rule[k, 1], Rule[z, Rational[1, 2]]}

Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[k, 2], Rule[z, Rational[1, 2]]}

... skip entries to safe data
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