# Fungrim entry: ed210c

Symbol: Log $\log(z)$ Natural logarithm
The principal branch of the natural logarithm $\log(z)$ is a function of one complex variable $z$.
It has a branch point singularity at $z = 0$ and a branch cut on $\left(-\infty, 0\right]$ where the value on $\left(-\infty, 0\right)$ is taken to be continuous with the upper half plane.
The following table lists all conditions such that Log(z) is defined in Fungrim.
Domain Codomain
Numbers
$z \in \left\{1\right\}$ $\log(z) \in \left\{0\right\}$
$z \in \left(0, \infty\right)$ $\log(z) \in \mathbb{R}$
$z \in \mathbb{C} \setminus \left\{0\right\}$ $\log(z) \in \mathbb{C}$
Infinities
$z \in \left\{\infty\right\}$ $\log(z) \in \left\{\infty\right\}$
Formal power series
$z \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z = 1$ $\log(z) \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] \log(z) = 0$
$z \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \in \left(0, \infty\right)$ $\log(z) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \ne 0$ $\log(z) \in \mathbb{C}[[x]]$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \;\implies\; \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
Log$\log(z)$ Natural logarithm
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
OpenInterval$\left(a, b\right)$ Open interval
RR$\mathbb{R}$ Real numbers
CC$\mathbb{C}$ Complex numbers
PowerSeries$K[[x]]$ Formal power series
QQ$\mathbb{Q}$ Rational numbers
Source code for this entry:
Entry(ID("ed210c"),
SymbolDefinition(Log, Log(z), "Natural logarithm"),
Description("The principal branch of the natural logarithm", Log(z), "is a function of one complex variable", z, "."),
Description("It has a branch point singularity at", Equal(z, 0), "and a branch cut on", OpenClosedInterval(Neg(Infinity), 0), "where the value on", OpenInterval(Neg(Infinity), 0), "is taken to be continuous with the upper half plane."),
Description("The following table lists all conditions such that", SourceForm(Log(z)), "is defined in Fungrim."),
Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, Set(1)), Element(Log(z), Set(0))), Tuple(Element(z, OpenInterval(0, Infinity)), Element(Log(z), RR)), Tuple(Element(z, SetMinus(CC, Set(0))), Element(Log(z), CC)), TableSection("Infinities"), Tuple(Element(z, Set(Infinity)), Element(Log(z), Set(Infinity))), TableSection("Formal power series"), Tuple(And(Element(z, PowerSeries(QQ, x)), Equal(SeriesCoefficient(z, x, 0), 1)), And(Element(Log(z), PowerSeries(QQ, x)), Equal(SeriesCoefficient(Log(z), x, 0), 0))), Tuple(And(Element(z, PowerSeries(RR, x)), Element(SeriesCoefficient(z, x, 0), OpenInterval(0, Infinity))), And(Element(Log(z), PowerSeries(RR, x)))), Tuple(And(Element(z, PowerSeries(CC, x)), NotEqual(SeriesCoefficient(z, x, 0), 0)), And(Element(Log(z), PowerSeries(CC, x)))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC