Elementary Functions - 4.4 Special Values and Limits

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.4.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@@{1} = 0}
\ln@@{1} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
ln(1) = 0

Log[1] == 0
Successful Successful - Successful [Tested: 1]
4.4.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@{-1+\iunit 0} = +\pi\iunit}
\ln@{-1+\iunit 0} = +\pi\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
ln(- 1 + I*0) = + Pi*I

Log[- 1 + I*0] == + Pi*I
Successful Successful Skip - symbolical successful subtest Successful [Tested: 1]
4.4.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@{-1-\iunit 0} = -\pi\iunit}
\ln@{-1-\iunit 0} = -\pi\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@{+\iunit} = +\tfrac{1}{2}\pi\iunit}
\ln@{+\iunit} = +\tfrac{1}{2}\pi\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
ln(+ I) = +(1)/(2)*Pi*I

Log[+ I] == +Divide[1,2]*Pi*I
Successful Successful - Successful [Tested: 1]
4.4.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@{-\iunit} = -\tfrac{1}{2}\pi\iunit}
\ln@{-\iunit} = -\tfrac{1}{2}\pi\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
ln(- I) = -(1)/(2)*Pi*I

Log[- I] == -Divide[1,2]*Pi*I
Successful Successful - Successful [Tested: 1]
4.4.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{0} = 1}
e^{0} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(0) = 1
 
Exp[0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
4.4.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+\pi\iunit} = -1}
e^{+\pi\iunit} = -1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(+ Pi*I) = - 1

Exp[+ Pi*I] == - 1
Successful Successful - Successful [Tested: 1]
4.4.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\pi\iunit} = -1}
e^{-\pi\iunit} = -1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(- Pi*I) = - 1

Exp[- Pi*I] == - 1
Successful Successful - Successful [Tested: 1]
4.4.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+\pi\iunit/2} = +\iunit}
e^{+\pi\iunit/2} = +\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(+ Pi*I/2) = + I

Exp[+ Pi*I/2] == + I
Successful Successful - Successful [Tested: 1]
4.4.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\pi\iunit/2} = -\iunit}
e^{-\pi\iunit/2} = -\iunit		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(- Pi*I/2) = - I

Exp[- Pi*I/2] == - I
Successful Successful - Successful [Tested: 1]
4.4.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{2\pi k\iunit} = 1}
e^{2\pi k\iunit} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(2*Pi*k*I) = 1

Exp[2*Pi*k*I] == 1
Successful Successful - Successful [Tested: 1]
4.4.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+\pi\iunit/3} = \frac{1}{2}+\iunit\frac{\sqrt{3}}{2}}
e^{+\pi\iunit/3} = \frac{1}{2}+\iunit\frac{\sqrt{3}}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\pi\iunit/3} = \frac{1}{2}-\iunit\frac{\sqrt{3}}{2}}
e^{-\pi\iunit/3} = \frac{1}{2}-\iunit\frac{\sqrt{3}}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+ 2\pi\iunit/3} = -\frac{1}{2}+\iunit\frac{\sqrt{3}}{2}}
e^{+ 2\pi\iunit/3} = -\frac{1}{2}+\iunit\frac{\sqrt{3}}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{- 2\pi\iunit/3} = -\frac{1}{2}-\iunit\frac{\sqrt{3}}{2}}
e^{- 2\pi\iunit/3} = -\frac{1}{2}-\iunit\frac{\sqrt{3}}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+\pi\iunit/4} = \frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}}
e^{+\pi\iunit/4} = \frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\pi\iunit/4} = \frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}}
e^{-\pi\iunit/4} = \frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+ 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}}
e^{+ 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}+\iunit\frac{1}{\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{- 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}}
e^{- 3\pi\iunit/4} = -\frac{1}{\sqrt{2}}-\iunit\frac{1}{\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \iunit^{+\iunit} = e^{-\pi/2}}
\iunit^{+\iunit} = e^{-\pi/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(I)^(+ I) = exp(- Pi/2)

(I)^(+ I) == Exp[- Pi/2]
Successful Successful - Successful [Tested: 1]
4.4.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \iunit^{-\iunit} = e^{+\pi/2}}
\iunit^{-\iunit} = e^{+\pi/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(I)^(- I) = exp(+ Pi/2)

(I)^(- I) == Exp[+ Pi/2]
Successful Successful - Successful [Tested: 1]
4.4.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{x\to\infty}x^{-a}\ln@@{x} = 0}
\lim_{x\to\infty}x^{-a}\ln@@{x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{x\to 0}x^{a}\ln@@{x} = 0}
\lim_{x\to 0}x^{a}\ln@@{x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{x\to\infty}x^{a}e^{-x} = 0}
\lim_{x\to\infty}x^{a}e^{-x} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{z\to\infty}z^{a}e^{-z} = 0}
\lim_{z\to\infty}z^{a}e^{-z} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln@@{n}\right) = \EulerConstant}
\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln@@{n}\right) = \EulerConstant		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerConstant = 0.57721\ 56649\ 01532\ 86060\dots}
\EulerConstant = 0.57721\ 56649\ 01532\ 86060\dots		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
gamma = 0.57721566490153286060

EulerGamma == 0.57721566490153286060
Successful Successful - Successful [Tested: 1]
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