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 ln ⁡ 1 = 0 1 0 {\displaystyle{\displaystyle\ln 1=0}}
\ln@@{1} = 0		 

ln(1) = 0

Log[1] == 0
Successful Successful - Successful [Tested: 1]
4.4.E2 ln ⁡ ( - 1 + i ⁢ 0 ) = + π ⁢ i 1 imaginary-unit 0 𝜋 imaginary-unit {\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 ln ⁡ ( - 1 - i ⁢ 0 ) = - π ⁢ i 1 imaginary-unit 0 𝜋 imaginary-unit {\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 ln ⁡ ( + i ) = + 1 2 ⁢ π ⁢ i imaginary-unit 1 2 𝜋 imaginary-unit {\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 ln ⁡ ( - i ) = - 1 2 ⁢ π ⁢ i imaginary-unit 1 2 𝜋 imaginary-unit {\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 e 0 = 1 superscript 𝑒 0 1 {\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 e + π ⁢ i = - 1 superscript 𝑒 𝜋 imaginary-unit 1 {\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 e - π ⁢ i = - 1 superscript 𝑒 𝜋 imaginary-unit 1 {\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 e + π ⁢ i / 2 = + i superscript 𝑒 𝜋 imaginary-unit 2 imaginary-unit {\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 e - π ⁢ i / 2 = - i superscript 𝑒 𝜋 imaginary-unit 2 imaginary-unit {\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 e 2 ⁢ π ⁢ k ⁢ i = 1 superscript 𝑒 2 𝜋 𝑘 imaginary-unit 1 {\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 e + π ⁢ i / 3 = 1 2 + i ⁢ 3 2 superscript 𝑒 𝜋 imaginary-unit 3 1 2 imaginary-unit 3 2 {\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 e - π ⁢ i / 3 = 1 2 - i ⁢ 3 2 superscript 𝑒 𝜋 imaginary-unit 3 1 2 imaginary-unit 3 2 {\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 e + 2 ⁢ π ⁢ i / 3 = - 1 2 + i ⁢ 3 2 superscript 𝑒 2 𝜋 imaginary-unit 3 1 2 imaginary-unit 3 2 {\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 e - 2 ⁢ π ⁢ i / 3 = - 1 2 - i ⁢ 3 2 superscript 𝑒 2 𝜋 imaginary-unit 3 1 2 imaginary-unit 3 2 {\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 e + π ⁢ i / 4 = 1 2 + i ⁢ 1 2 superscript 𝑒 𝜋 imaginary-unit 4 1 2 imaginary-unit 1 2 {\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 e - π ⁢ i / 4 = 1 2 - i ⁢ 1 2 superscript 𝑒 𝜋 imaginary-unit 4 1 2 imaginary-unit 1 2 {\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 e + 3 ⁢ π ⁢ i / 4 = - 1 2 + i ⁢ 1 2 superscript 𝑒 3 𝜋 imaginary-unit 4 1 2 imaginary-unit 1 2 {\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 e - 3 ⁢ π ⁢ i / 4 = - 1 2 - i ⁢ 1 2 superscript 𝑒 3 𝜋 imaginary-unit 4 1 2 imaginary-unit 1 2 {\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 i + i = e - π / 2 imaginary-unit imaginary-unit superscript 𝑒 𝜋 2 {\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 i - i = e + π / 2 imaginary-unit imaginary-unit superscript 𝑒 𝜋 2 {\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 lim x → ∞ ⁡ x - a ⁢ ln ⁡ x = 0 subscript → 𝑥 superscript 𝑥 𝑎 𝑥 0 {\displaystyle{\displaystyle\lim_{x\to\infty}x^{-a}\ln x=0}}
\lim_{x\to\infty}x^{-a}\ln@@{x} = 0		 ℜ ⁡ a > 0 𝑎 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 lim x → 0 ⁡ x a ⁢ ln ⁡ x = 0 subscript → 𝑥 0 superscript 𝑥 𝑎 𝑥 0 {\displaystyle{\displaystyle\lim_{x\to 0}x^{a}\ln x=0}}
\lim_{x\to 0}x^{a}\ln@@{x} = 0		 ℜ ⁡ a > 0 𝑎 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 lim x → ∞ ⁡ x a ⁢ e - x = 0 subscript → 𝑥 superscript 𝑥 𝑎 superscript 𝑒 𝑥 0 {\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 lim z → ∞ ⁡ z a ⁢ e - z = 0 subscript → 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 0 {\displaystyle{\displaystyle\lim_{z\to\infty}z^{a}e^{-z}=0}}
\lim_{z\to\infty}z^{a}e^{-z} = 0		 | ph ⁡ z | ≤ 1 2 ⁢ π - δ , 1 2 ⁢ π - δ < 1 2 ⁢ π formulae-sequence phase 𝑧 1 2 𝜋 𝛿 1 2 𝜋 𝛿 1 2 𝜋 {\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 lim n → ∞ ⁡ ( 1 + z n ) n = e z subscript → 𝑛 superscript 1 𝑧 𝑛 𝑛 superscript 𝑒 𝑧 {\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}		 z = 𝑧 absent {\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 lim n → ∞ ⁡ ( 1 + 1 n ) n = e subscript → 𝑛 superscript 1 1 𝑛 𝑛 𝑒 {\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 lim n → ∞ ⁡ ( ( ∑ k = 1 n 1 k ) - ln ⁡ n ) = γ subscript → 𝑛 subscript superscript 𝑛 𝑘 1 1 𝑘 𝑛 {\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 γ = 0.57721 56649 01532 86060 ⁢ … 0.57721 56649 01532 86060 … {\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]
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