4.4

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.4.E1	${\displaystyle{\displaystyle\ln 1=0}}$ \ln@@{1} = 0		ln(1) = 0	Log[1] == 0	Successful	Successful	-	Successful [Tested: 1]
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1+\mathrm{i}0\right)=+\pi\mathrm{i}}}$ \ln@{-1+\iunit 0} = +\pi\iunit		ln(- 1 + I*0) = + Pi*I	Log[- 1 + I*0] == + Pi*I	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 1]
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1-\mathrm{i}0\right)=-\pi\mathrm{i}}}$ \ln@{-1-\iunit 0} = -\pi\iunit		ln(- 1 - I*0) = - Pi*I	Log[- 1 - I*0] == - Pi*I	Failure	Failure	Failed [1 / 1] Result: 6.283185308*I Test Values: {}	Failed [1 / 1] Result: Complex[0.0, 6.283185307179586] Test Values: {}
4.4.E3	${\displaystyle{\displaystyle\ln\left(+\mathrm{i}\right)=+\tfrac{1}{2}\pi\mathrm{i}}}$ \ln@{+\iunit} = +\tfrac{1}{2}\pi\iunit		ln(+ I) = +(1)/(2)*Pi*I	Log[+ I] == +Divide[1,2]*Pi*I	Successful	Successful	-	Successful [Tested: 1]
4.4.E3	${\displaystyle{\displaystyle\ln\left(-\mathrm{i}\right)=-\tfrac{1}{2}\pi\mathrm{i}}}$ \ln@{-\iunit} = -\tfrac{1}{2}\pi\iunit		ln(- I) = -(1)/(2)*Pi*I	Log[- I] == -Divide[1,2]*Pi*I	Successful	Successful	-	Successful [Tested: 1]
4.4.E4	${\displaystyle{\displaystyle e^{0}=1}}$ e^{0} = 1		exp(0) = 1	Exp[0] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E5	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}}=-1}}$ e^{+\pi\iunit} = -1		exp(+ Pi*I) = - 1	Exp[+ Pi*I] == - 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E5	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}}=-1}}$ e^{-\pi\iunit} = -1		exp(- Pi*I) = - 1	Exp[- Pi*I] == - 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E6	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/2}=+\mathrm{i}}}$ e^{+\pi\iunit/2} = +\iunit		exp(+ Pi*I/2) = + I	Exp[+ Pi*I/2] == + I	Successful	Successful	-	Successful [Tested: 1]
4.4.E6	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/2}=-\mathrm{i}}}$ e^{-\pi\iunit/2} = -\iunit		exp(- Pi*I/2) = - I	Exp[- Pi*I/2] == - I	Successful	Successful	-	Successful [Tested: 1]
4.4.E7	${\displaystyle{\displaystyle e^{2\pi k\mathrm{i}}=1}}$ e^{2\pi k\iunit} = 1		exp(2*Pi*k*I) = 1	Exp[2*Pi*k*I] == 1	Successful	Successful	-	Successful [Tested: 1]
4.4.E8	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/3}=\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2}}}$ e^{+\pi\iunit/3} = \frac{1}{2}+\iunit\frac{\sqrt{3}}{2}		exp(+ Pi*I/3) = (1)/(2)+ I*(sqrt(3))/(2)	Exp[+ Pi*I/3] == Divide[1,2]+ I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E8	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/3}=\frac{1}{2}-\mathrm{i}\frac{\sqrt{3}}{2}}}$ e^{-\pi\iunit/3} = \frac{1}{2}-\iunit\frac{\sqrt{3}}{2}		exp(- Pi*I/3) = (1)/(2)- I*(sqrt(3))/(2)	Exp[- Pi*I/3] == Divide[1,2]- I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E9	${\displaystyle{\displaystyle e^{+2\pi\mathrm{i}/3}=-\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2}}}$ e^{+ 2\pi\iunit/3} = -\frac{1}{2}+\iunit\frac{\sqrt{3}}{2}		exp(+ 2*Pi*I/3) = -(1)/(2)+ I*(sqrt(3))/(2)	Exp[+ 2*Pi*I/3] == -Divide[1,2]+ I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E9	${\displaystyle{\displaystyle e^{-2\pi\mathrm{i}/3}=-\frac{1}{2}-\mathrm{i}\frac{\sqrt{3}}{2}}}$ e^{- 2\pi\iunit/3} = -\frac{1}{2}-\iunit\frac{\sqrt{3}}{2}		exp(- 2*Pi*I/3) = -(1)/(2)- I*(sqrt(3))/(2)	Exp[- 2*Pi*I/3] == -Divide[1,2]- I*Divide[Sqrt[3],2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E10	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}+\mathrm{i}\frac{1}{\sqrt{2}}}}$ e^{+\pi\iunit/4} = \frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}		exp(+ Pi*I/4) = (1)/(sqrt(2))+ I*(1)/(sqrt(2))	Exp[+ Pi*I/4] == Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E10	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}-\mathrm{i}\frac{1}{\sqrt{2}}}}$ e^{-\pi\iunit/4} = \frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}		exp(- Pi*I/4) = (1)/(sqrt(2))- I*(1)/(sqrt(2))	Exp[- Pi*I/4] == Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E11	${\displaystyle{\displaystyle e^{+3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}+\mathrm{i}\frac{1}{\sqrt{2}}}}$ e^{+ 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}		exp(+ 3*Pi*I/4) = -(1)/(sqrt(2))+ I*(1)/(sqrt(2))	Exp[+ 3*Pi*I/4] == -Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E11	${\displaystyle{\displaystyle e^{-3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}-\mathrm{i}\frac{1}{\sqrt{2}}}}$ e^{- 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}		exp(- 3*Pi*I/4) = -(1)/(sqrt(2))- I*(1)/(sqrt(2))	Exp[- 3*Pi*I/4] == -Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]]	Successful	Successful	-	Successful [Tested: 1]
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{+\mathrm{i}}}=e^{-\pi/2}}}$ \iunit^{+\iunit} = e^{-\pi/2}		(I)^(+ I) = exp(- Pi/2)	(I)^(+ I) == Exp[- Pi/2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{-\mathrm{i}}}=e^{+\pi/2}}}$ \iunit^{-\iunit} = e^{+\pi/2}		(I)^(- I) = exp(+ Pi/2)	(I)^(- I) == Exp[+ Pi/2]	Successful	Successful	-	Successful [Tested: 1]
4.4.E13	${\displaystyle{\displaystyle\lim_{x\to\infty}x^{-a}\ln x=0}}$ \lim_{x\to\infty}x^{-a}\ln@@{x} = 0	${\displaystyle{\displaystyle\Re a>0}}$	limit((x)^(- a)* ln(x), x = infinity) = 0	Limit[(x)^(- a)* Log[x], x -> Infinity, GenerateConditions->None] == 0	Successful	Successful	-	Successful [Tested: 3]
4.4.E14	${\displaystyle{\displaystyle\lim_{x\to 0}x^{a}\ln x=0}}$ \lim_{x\to 0}x^{a}\ln@@{x} = 0	${\displaystyle{\displaystyle\Re a>0}}$	limit((x)^(a)* ln(x), x = 0) = 0	Limit[(x)^(a)* Log[x], x -> 0, GenerateConditions->None] == 0	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.4.E15	${\displaystyle{\displaystyle\lim_{x\to\infty}x^{a}e^{-x}=0}}$ \lim_{x\to\infty}x^{a}e^{-x} = 0		limit((x)^(a)* exp(- x), x = infinity) = 0	Limit[(x)^(a)* Exp[- x], x -> Infinity, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E16	${\displaystyle{\displaystyle\lim_{z\to\infty}z^{a}e^{-z}=0}}$ \lim_{z\to\infty}z^{a}e^{-z} = 0	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta,\tfrac{1}{2}\pi-\delta<\tfrac{1}{2}\pi}}$	limit((z)^(a)* exp(- z), z = infinity) = 0	Limit[(z)^(a)* Exp[- z], z -> Infinity, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E17	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n}=e^{z}}}$ \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n} = e^{z}	${\displaystyle{\displaystyle z=}}$	limit((1 +(z)/(n))^(n), n = infinity) = exp(z)	Limit[(1 +Divide[z,n])^(n), n -> Infinity, GenerateConditions->None] == Exp[z]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E18	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e}}$ \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = e		limit((1 +(1)/(n))^(n), n = infinity) = exp(1)	Limit[(1 +Divide[1,n])^(n), n -> Infinity, GenerateConditions->None] == E	Skipped - no semantic math	Skipped - no semantic math	-	-
4.4.E19	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln n\right)=\gamma}}$ \lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln@@{n}\right) = \EulerConstant		limit((sum((1)/(k), k = 1..n))- ln(n), n = infinity) = gamma	Limit[(Sum[Divide[1,k], {k, 1, n}, GenerateConditions->None])- Log[n], n -> Infinity, GenerateConditions->None] == EulerGamma	Successful	Successful	-	Successful [Tested: 1]
4.4.E19	${\displaystyle{\displaystyle\gamma=0.57721\ 56649\ 01532\ 86060\dots}}$ \EulerConstant = 0.57721\ 56649\ 01532\ 86060\dots		gamma = 0.57721566490153286060	EulerGamma == 0.57721566490153286060	Successful	Successful	-	Successful [Tested: 1]