Log

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

log(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Quasi-homogeneity

Scale symmetry

Periodicity

No periodicity

Poles and essential singularities

The function log(z) does not have poles and essential singularities.

Branch points

Branch cuts

Analytic continuations

log(z) is chosen to be the principal branch of the general logarithmic function that has infinitely many sheets, each differing from log(z) by 2π i k, k∈ℤ.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==1

Expansions of Log[1+z] at z==0

Expansions of Log[1+z] at z==∞

Asymptotic series expansions

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Product representations

Ruffa Anthony

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving complex characteristics

Argument involving basic arithmetic operations

Addition formulas

Half-angle formulas

Multiple arguments

For products

For quotients

Power of arguments

Exponent of arguments

Some functions of arguments

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Differences of the direct function

Linear combinations of the direct function

Related transformations

Sums involving the direct function

Differences involving the direct function

Linear combinations involving the direct function

Identities

Functional identities

Univariate functional identities

Bivariate functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

For the direct function itself

Involving the direct function

Definite integration

For the direct function itself

Involving the direct function

Involving related functions

Double integrals

Integral transforms

Laplace transforms

Summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving algebraic functions in the arguments

Classical cases involving powers of Log involving algebraic functions

Classical cases for powers of Log involving algebraic functions in the arguments

Classical cases involving unit step UnitStep

Classical cases for powers of Log involving unit step UnitStep

Classical cases involving algebraic functions in the arguments and unit step UnitStep

Classical cases involving Abs in the arguments

Classical cases involving Sign in the arguments

Generalized cases for the direct function itself

Generalized cases involving algebraic functions in the arguments

Generalized cases for powers of Log involving algebraic functions in the arguments

Generalized cases involving algebraic functions in the arguments and unit step UnitStep

Generalized cases involving Sign in the arguments

Through other functions

Representations through equivalent functions

With inverse function Log

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==log(z), which generically does not equal to z.

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==log(z), which equal to z under restriction -π<Im(z)≤π.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==log(z), which generically equal to z.

With related functions

Involving ArcSin

Involving ArcCos

Involving ArcTan

Involving ArcCot

Involving ArcCsc

Involving ArcSec

Involving ArcSinh

Involving ArcCosh

Involving ArcTanh

Involving ArcCoth

Involving ArcCsch

Involving ArcSech

Involving Log[a, z]

Involving other related functions

Inequalities

Zeros

Theorems

The Shannon information content of a message

The Shannon information content of a message is H==-(∑_i)(Subscript[p, i]) log((Subscript[p, i])) , where (Subscript[p, i]) is the probability for the symbol (Subscript[ξ, i]).

The entropy of a (classical) physical system

The entropy S of a (classical) physical system is S==(Subscript[k, B]) log(Γ(E)), where (Subscript[k, B]) is the Boltzmann constant and Γ(E) is the phase space volume occupied by the system at energy E.

Transcendentality of a sum

If α==(∑_(i=1)^n)(Subscript[γ, i])log((Subscript[β, i]))≠0 , where the (Subscript[β, i]), (Subscript[γ, i]) are algebraic numbers, then α is transcendental.

Conformal mapping from the half-plane to the strip

The conformal map from the upper half w -plane to the strip -∞<Im(z)<0,0<Re(z)<π is given by z(w)==log(w)+(1-(w^2))/2.

Differential-Algebraic Constants

To ensure the correctness of many formulas given in this collection over the whole complex plane, it is often necessary to work with expressions of the form log((z^2))==log(i z)+log(-i z) , log((z^2))-2log(z), log(z)+log(1/z), log(-(z^2))==log(z)+log(-z), etc.. While in a textbook-mathematics setting these expressions are often simplified to 2log(z), 0, 0,  2log(z)+π i, etc, this cannot be done inside Mathematica. From a complex function point of view the Riemann surface of such functions are made from disconnected sheets. Inside Mathematica all branch cuts of all functions (that have branch cuts) follow uniquely from the branch cut of the power function (the logarithm function respectively). As a result the branch cuts related to functions such as  log((z^2)), log((z^2))-2log(z), log(z)+log(1/z), log(-(z^2)) exist, although they do not start and end at branch points.
In details we have:

(Subscript[ℬ𝒞, z])(log((z^2)))==(Subscript[ℬ𝒞, z])(log(i z)+log(-i z))=={{(-i ∞,0),1},{(0,i ∞),-1}}

(Subscript[ℬ𝒞, z])(log((z^2))-2log(z))=={{(-i ∞,0),1},{(0,i ∞),-1},{(-∞,0),-i}}

(Subscript[ℬ𝒞, z])(log(z)+log(1/z))=={{(-∞,0),{}}}

An expression of the form log((z^2))-2log(z) , log(z)+log(1/z) are called differential-algebraic constants because their derivative vanishes generically everywhere as a complex function (but not as a generalized function).

History

– J. Napier (1614) published the first tables and used word Log
– H. Briggs (1617) published the first tables in base 10 and found logarithms of the first 25 primes
– J. Burgi (1620)
– J. Kepler (1624)
– B. Cavalieri (1632)
– J. Gregory (1668) found series expansion for log
– N. Mercator (1668) used "Log naturalis"
– J. N. Lambert (1770) and J.-L. Lagrange (1776) found continued fraction representations for Log
– L. Euler (1749) found that log was multivalued

The function log is encountered often in mathematics and the natural sciences.

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