4.8

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.8.E1	${\displaystyle{\displaystyle\operatorname{Ln}\left(z_{1}z_{2}\right)=\operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}}}$ \Ln@{z_{1}z_{2}} = \Ln@@{z_{1}}+\Ln@@{z_{2}}		ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])	Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]	Failure	Failure	Failed [25 / 100] Result: 0.-6.283185308*I Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -1.5} Result: 0.-6.283185308*I Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -.5} Result: -.1e-9-6.283185308*I Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -2} Result: .133199999e-10-6.283185307*I Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [25 / 100] Result: Complex[0.0, -6.283185307179587] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -1.5]} Result: Complex[0.0, -6.283185307179587] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -0.5]} ... skip entries to safe data
4.8.E2	${\displaystyle{\displaystyle\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2}}}$ \ln@{z_{1}z_{2}} = \ln@@{z_{1}}+\ln@@{z_{2}}	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}+\operatorname{ph}z_{2},\operatorname{ph}z_{1}+\operatorname{ph}z_{2}\leq\pi}}$	ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])	Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]	Failure	Failure	Successful [Tested: 59]	Successful [Tested: 75]
4.8.E3	${\displaystyle{\displaystyle\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname{Ln}z_{1}-\operatorname{Ln}z_{2}}}$ \Ln@@{\frac{z_{1}}{z_{2}}} = \Ln@@{z_{1}}-\Ln@@{z_{2}}		ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])	Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]	Failure	Failure	Failed [25 / 100] Result: 0.-6.283185307*I Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2*3^(1/2)-1/2*I} Result: 0.+6.283185307*I Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)} Result: .1e-9+6.283185307*I Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1.5} Result: -.1e-9+6.283185307*I Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -.5} ... skip entries to safe data	Failed [25 / 100] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E4	${\displaystyle{\displaystyle\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2}}}$ \ln@@{\frac{z_{1}}{z_{2}}} = \ln@@{z_{1}}-\ln@@{z_{2}}	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}-\operatorname{ph}z_{2},\operatorname{ph}z_{1}-\operatorname{ph}z_{2}\leq\pi}}$	ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])	Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]	Failure	Failure	Failed [3 / 70] Result: 0.+6.283185307*I Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)} Result: 0.+6.283185308*I Test Values: {z[1] = -1/2*3^(1/2)-1/2*I, z[2] = 1/2*3^(1/2)+1/2*I} Result: 6.283185308*I Test Values: {z[1] = 2, z[2] = -2}	Failed [11 / 86] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.8.E5	${\displaystyle{\displaystyle\operatorname{Ln}\left(z^{n}\right)=n\operatorname{Ln}z}}$ \Ln@{z^{n}} = n\Ln@@{z}		ln((z)^(n)) = n*ln(z)	Log[(z)^(n)] == n*Log[z]	Failure	Failure	Failed [5 / 21] Result: .133199999e-10-6.283185307*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2, n = 3} Result: .4399599996e-9-6.283185306*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3, n = 3} Result: .4399599996e-9+6.283185306*I Test Values: {z = 1/2-1/2*I*3^(1/2), n = 3, n = 3} Result: .133199999e-10+6.283185307*I Test Values: {z = -1/2*3^(1/2)-1/2*I, n = 2, n = 3} ... skip entries to safe data	Failed [3 / 7] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.8.E6	${\displaystyle{\displaystyle\ln\left(z^{n}\right)=n\ln z}}$ \ln@{z^{n}} = n\ln@@{z}	${\displaystyle{\displaystyle-\pi\leq n\operatorname{ph}z,n\operatorname{ph}z\leq\pi}}$	ln((z)^(n)) = n*ln(z)	Log[(z)^(n)] == n*Log[z]	Failure	Failure	Failed [1 / 17] Result: .4399599996e-9+6.283185306*I Test Values: {z = 1/2-1/2*I*3^(1/2), n = 3, n = 3}	Failed [3 / 7] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.8.E7	${\displaystyle{\displaystyle\ln\frac{1}{z}=-\ln z}}$ \ln@@{\frac{1}{z}} = -\ln@@{z}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi}}$	ln((1)/(z)) = - ln(z)	Log[Divide[1,z]] == - Log[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.8.E8	${\displaystyle{\displaystyle\operatorname{Ln}\left(\exp z\right)=z+2k\pi\mathrm{i}}}$ \Ln@{\exp@@{z}} = z+2k\pi\iunit		ln(exp(z)) = z + 2*k*Pi*I	Log[Exp[z]] == z + 2*k*Pi*I	Failure	Failure	Failed [21 / 21] Result: -.1e-9-6.283185308*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3} Result: -.1e-9-12.56637062*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3} Result: -.1e-9-18.84955592*I Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3} Result: 0.-6.283185308*I Test Values: {z = -1/2+1/2*I*3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E9	${\displaystyle{\displaystyle\ln\left(\exp z\right)=z}}$ \ln@{\exp@@{z}} = z	${\displaystyle{\displaystyle-\pi\leq\Im z,\Im z\leq\pi}}$	ln(exp(z)) = z	Log[Exp[z]] == z	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.8.E10	${\displaystyle{\displaystyle\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}z\right)}}$ \exp@{\ln@@{z}} = \exp@{\Ln@@{z}}		exp(ln(z)) = exp(ln(z))	Exp[Log[z]] == Exp[Log[z]]	Successful	Successful	-	Successful [Tested: 7]
4.8.E10	${\displaystyle{\displaystyle\exp\left(\operatorname{Ln}z\right)=z}}$ \exp@{\Ln@@{z}} = z		exp(ln(z)) = z	Exp[Log[z]] == z	Successful	Successful	-	Successful [Tested: 7]
4.8.E11	${\displaystyle{\displaystyle\operatorname{Ln}\left(a^{z}\right)=z\operatorname{Ln}a+2k\pi\mathrm{i}}}$ \Ln@{a^{z}} = z\Ln@@{a}+2k\pi\iunit		ln((a)^(z)) = z*ln(a)+ 2*k*Pi*I	Log[(a)^(z)] == z*Log[a]+ 2*k*Pi*I	Failure	Failure	Failed [126 / 126] Result: 0.-6.283185308*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3} Result: 0.-12.56637062*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3} Result: 0.-18.84955592*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3} Result: 0.-6.283185308*I Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2), k = 1, k = 3} ... skip entries to safe data	Failed [42 / 42] Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -18.84955592153876] Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.8.E12	${\displaystyle{\displaystyle\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i}}}$ \ln@{a^{z}} = z\ln@@{a}+2k\pi\iunit		ln((a)^(z)) = z*ln(a)+ 2*k*Pi*I	Log[(a)^(z)] == z*Log[a]+ 2*k*Pi*I	Failure	Failure	Failed [126 / 126] Result: 0.-6.283185308*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 1} Result: 0.-12.56637062*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 2} Result: 0.-18.84955592*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 3} Result: 0.-6.283185308*I Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2), k = 1} ... skip entries to safe data	Failed [126 / 126] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -12.566370614359172] Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.8.E13	${\displaystyle{\displaystyle\ln\left(a^{x}\right)=x\ln a}}$ \ln@{a^{x}} = x\ln@@{a}	${\displaystyle{\displaystyle a>0}}$	ln((a)^(x)) = x*ln(a)	Log[(a)^(x)] == x*Log[a]	Successful	Failure	-	Successful [Tested: 9]
4.8.E14	${\displaystyle{\displaystyle a^{z_{1}}a^{z_{2}}=a^{z_{1}+z_{2}}}}$ a^{z_{1}}a^{z_{2}} = a^{z_{1}+z_{2}}	${\displaystyle{\displaystyle a\neq 0}}$	(a)^(z[1])* (a)^(z[2]) = (a)^(z[1]+ z[2])	(a)^(Subscript[z, 1])* (a)^(Subscript[z, 2]) == (a)^(Subscript[z, 1]+ Subscript[z, 2])	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E15	${\displaystyle{\displaystyle a^{z}b^{z}=(ab)^{z}}}$ a^{z}b^{z} = (ab)^{z}	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}a+\operatorname{ph}b,\operatorname{ph}a+\operatorname{ph}b\leq\pi}}$	(a)^(z)* (b)^(z) = (a*b)^(z)	(a)^(z)* (b)^(z) == (a*b)^(z)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E16	${\displaystyle{\displaystyle e^{z_{1}}e^{z_{2}}=e^{z_{1}+z_{2}}}}$ e^{z_{1}}e^{z_{2}} = e^{z_{1}+z_{2}}		exp(z[1])*exp(z[2]) = exp(z[1]+ z[2])	Exp[Subscript[z, 1]]*Exp[Subscript[z, 2]] == Exp[Subscript[z, 1]+ Subscript[z, 2]]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.8.E17	${\displaystyle{\displaystyle(e^{z_{1}})^{z_{2}}=e^{z_{1}z_{2}}}}$ (e^{z_{1}})^{z_{2}} = e^{z_{1}z_{2}}	${\displaystyle{\displaystyle-\pi\leq\Im z_{1},\Im z_{1}\leq\pi}}$	(exp(z[1]))^(z[2]) = exp(z[1]*z[2])	(Exp[Subscript[z, 1]])^(Subscript[z, 2]) == Exp[Subscript[z, 1]*Subscript[z, 2]]	Skipped - no semantic math	Skipped - no semantic math	-	-