Difference between revisions of "Logarithm"

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__NOTOC__
 
The (single-valued) logarithm $\log \colon \mathbb{C} \setminus (-\infty,0] \rightarrow \mathbb{C}$ defined by the formula
 
The (single-valued) logarithm $\log \colon \mathbb{C} \setminus (-\infty,0] \rightarrow \mathbb{C}$ defined by the formula
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$$\log(z) = \displaystyle\int_1^z \dfrac{1}{t} dt,$$
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$$\log(z) = \displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$
 
where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not intersect the set $(-\infty,0] \subset \mathbb{C}$.
 
where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not intersect the set $(-\infty,0] \subset \mathbb{C}$.
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The logarithm restricted to $(0,\infty)$ is the [[inverse function]] of the [[exponential function]] restricted to $\mathbb{R}$.
 
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
 
File:Logarithmplot.png|Graph of $\log$ on $(0,10]$.
 
File:Logarithmplot.png|Graph of $\log$ on $(0,10]$.
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File:Domcollogarithm.png|[[Domain coloring]] of $\log$ with [[branch cut]] $(-\infty,0]$.
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File:Complexlogarithm.png|[[Domain coloring]] of $\log$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
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=Properties=
 
=Properties=
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[[Logarithm of a complex number]]<br />
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[[Derivative of the logarithm]]<br />
 +
[[Real and imaginary parts of log]]<br />
 +
[[Relationship between logarithm (multivalued) and logarithm]]<br />
 +
[[Logarithm of product is a sum of logarithms]]<br />
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[[Logarithm of a quotient is a difference of logarithms]]<br />
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[[Relationship between logarithm and positive integer exponents]]<br />
 +
[[Logarithm of 1]]<br />
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[[Logarithm diverges to negative infinity at 0 from right]]<br />
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[[Logarithm at minus 1]]<br />
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[[Logarithm at i]]<br />
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[[Logarithm at -i]]<br />
 
[[Taylor series of log(1-z)|Taylor series of $\log(1-z)$]] <br />
 
[[Taylor series of log(1-z)|Taylor series of $\log(1-z)$]] <br />
 
[[Taylor series of log(1+z)|Taylor series of $\log(1+z)$]]<br />
 
[[Taylor series of log(1+z)|Taylor series of $\log(1+z)$]]<br />
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Antiderivative of the logarithm]]<br />
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<strong>Proposition:</strong> $\displaystyle\int \log(z) dz = z \log(z)-z$
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[[Z2F1(1,1;2,-z) equals log(1+z)]]<br />
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<div class="mw-collapsible-content">
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[[Exponential integral Ei series]]<br />
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<strong>Proof:</strong> █
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[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
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</div>
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[[Prime number theorem, pi and x/log(x)|The prime number theorem]]<br />
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</div>
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[[Series for log(z) for Re(z) greater than 1/2]]<br />
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[[Series for log(z) for absolute value of (z-1) less than 1]]<br />
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[[Series for log(z) for Re(z) greater than 0]]<br />
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[[Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1]]<br />
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[[Series for log(z+a) for positive a and Re(z) greater than -a]]<br />
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[[Relationship between logarithm and Mangoldt]]<br />
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[[Log e(z)=log(z)]]<br />
 +
[[Log 10(z)=log(z)/log(10)]]<br />
 +
[[Log 10(z)=log 10(e)log(z)]]<br />
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[[Log(z)=log(10)log 10(z)]]<br />
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[[Limit of log(x)/x^a=0]]<br />
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[[Limit of x^a log(x)=0]]<br />
 +
[[X/(1+x) less than log(1+x)]]<br />
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[[Log(1+x) less than x]]<br />
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[[X less than -log(1-x)]]<br />
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[[-log(1-x) less than x/(1-x)]]<br />
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[[Abs(log(1-x)) less than 3x/2]]<br />
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[[Log(x) less than or equal to x-1]]<br />
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[[Log(x) less than or equal to n(x^(1/n)-1)]]<br />
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[[Abs(log(1+z)) less than or equal to -log(1-abs(z))]]<br />
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[[Log(1+z) as continued fraction]]<br />
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[[Log((1+z)/(1-z)) as continued fraction]]<br />
  
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<div class="toccolours mw-collapsible mw-collapsed">
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=References=
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<strong>Theorem:</strong> For $|z|<1$,
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Continued fraction|next=Real and imaginary parts of log}}: $4.1.1$
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$$\log(1+z) = -\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k z^k}{k}.$$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Nth derivative of logarithm|next=Antiderivative of the logarithm}}: $4.1.48$
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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=Relation to other special functions=
 
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{{:Z2F1(1,1;2,-z) equals log(1+z)}}
 
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{{:Exponential integral Ei series}}
 
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{{:Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta}}
 
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=Major theorems involving the logarithm=
 
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[[Prime number theorem, pi and x/log(x)|The prime number theorem]]
 
  
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=References=
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{{:Logarithm and friends footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 05:03, 21 December 2017

The (single-valued) logarithm $\log \colon \mathbb{C} \setminus (-\infty,0] \rightarrow \mathbb{C}$ defined by the formula $$\log(z) = \displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not intersect the set $(-\infty,0] \subset \mathbb{C}$.


Properties

Logarithm of a complex number
Derivative of the logarithm
Real and imaginary parts of log
Relationship between logarithm (multivalued) and logarithm
Logarithm of product is a sum of logarithms
Logarithm of a quotient is a difference of logarithms
Relationship between logarithm and positive integer exponents
Logarithm of 1
Logarithm diverges to negative infinity at 0 from right
Logarithm at minus 1
Logarithm at i
Logarithm at -i
Taylor series of $\log(1-z)$
Taylor series of $\log(1+z)$
Antiderivative of the logarithm
Z2F1(1,1;2,-z) equals log(1+z)
Exponential integral Ei series
Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta
The prime number theorem
Series for log(z) for Re(z) greater than 1/2
Series for log(z) for absolute value of (z-1) less than 1
Series for log(z) for Re(z) greater than 0
Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1
Series for log(z+a) for positive a and Re(z) greater than -a
Relationship between logarithm and Mangoldt
Log e(z)=log(z)
Log 10(z)=log(z)/log(10)
Log 10(z)=log 10(e)log(z)
Log(z)=log(10)log 10(z)
Limit of log(x)/x^a=0
Limit of x^a log(x)=0
X/(1+x) less than log(1+x)
Log(1+x) less than x
X less than -log(1-x)
-log(1-x) less than x/(1-x)
Abs(log(1-x)) less than 3x/2
Log(x) less than or equal to x-1
Log(x) less than or equal to n(x^(1/n)-1)
Abs(log(1+z)) less than or equal to -log(1-abs(z))
Log(1+z) as continued fraction
Log((1+z)/(1-z)) as continued fraction

References

Logarithm and friends
Dilogarithmthumb.png
Dilogarithm
Lithumb.png
Log integral
Logthumb.png
Logarithm
Polylog1thumb.png
Polylogarithm
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