Fungrim symbol: Log

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 symbol

The symbol Log appears in 39 topics:

Natural logarithm (in 33 entries)
Barnes G-function (in 20 entries)
Digamma function (in 17 entries)
Specific values of Carlson symmetric elliptic integrals (in 15 entries)
Specific values of the digamma function (in 9 entries)
Riemann hypothesis (in 8 entries)
Catalan's constant (in 8 entries)
Carlson symmetric elliptic integrals (in 8 entries)
Prime numbers (in 7 entries)
Inverse tangent (in 7 entries)
Euler's constant (in 7 entries)
Fibonacci numbers (in 6 entries)
Totient function (in 5 entries)
Stieltjes constants (in 5 entries)
Pi (in 5 entries)
Landau's function (in 5 entries)
Powers (in 4 entries)
Legendre elliptic integrals (in 4 entries)
Lambert W-function (in 4 entries)
Keiper-Li coefficients (in 4 entries)
Integrals of Jacobi theta functions (in 4 entries)
Hurwitz zeta function (in 4 entries)
Gamma function (in 4 entries)
Bell numbers (in 3 entries)
Sinc function (in 2 entries)
Series and product representations of Jacobi theta functions (in 2 entries)
Exponential function (in 2 entries)
Bounds and inequalities for the gamma function (in 2 entries)
Stirling numbers (in 1 entries)
Square roots (in 1 entries)
Imaginary unit (in 1 entries)
Halphen's constant (in 1 entries)
Gauss hypergeometric function (in 1 entries)
Dirichlet characters (in 1 entries)
Dirichlet L-functions (in 1 entries)
Complex parts (in 1 entries)
Chebyshev polynomials (in 1 entries)
Bessel functions (in 1 entries)
Arithmetic-geometric mean (in 1 entries)

Entries using this symbol

The symbol Log appears in 202 entries: