Definition:Natural Logarithm/Complex

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Definition

Definition 1

Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.

The complex natural logarithm of $z \in \C_{\ne 0}$ is the multifunction defined as:

$\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$

where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.


Definition 2

Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The complex natural logarithm of $z$ is the multifunction defined as:

$\map \ln z := \set {w \in \C: e^w = z}$


Principal Branch

The principal branch of the complex natural logarithm is usually defined in one of two ways:

$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$
$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointl {-\pi} \pi$

It is important to specify which is in force during a particular exposition.


Notation

The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as:

$\ln z$
$\log z$
$\Log z$
$\log_e z$

The first of these is commonly encountered, and is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated.

The second and third are ambiguous (it doesn't tell you which base it is the logarithm of).

While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.


Examples

Logarithm of $-1$

$\map \ln {-1} = \paren {2 k + 1} \pi i$

for all $k \in \Z$.


Logarithm of $-2$

$\ln \paren {-2} = \ln 2 + \paren {2 k + 1} \pi i$

for all $k \in \Z$.


Logarithm of $i$

$\ln \paren i = \paren {4 k + 1} \dfrac {\pi i} 2$

for all $k \in \Z$.


Logarithm of $1 - i \tan \alpha$

$\ln \paren {1 - i \tan \alpha} = \ln \sec \alpha + i \paren {-\alpha + 2 k \pi}$

for all $k \in \Z$.


Also see

  • Results about logarithms can be found here.


Historical Note

In $1702$, Johann Bernoulli encountered solutions of the primitive $\displaystyle \int \dfrac {\d x} {a x^2 + b x + c}$ which seemed to require logarithms of complex numbers, which at that time had not been considered.

Johann Bernoulli and Gottfried Wilhelm von Leibniz both investigated, and by $1712$ they had developed opposing viewpoints on how to handle the logarithm of a negative number.

Bernoulli used the argument:

$\dfrac {\map d {-x} } {-x} = \dfrac {\map d x} x$, so by integration $\map \ln {-x} = \map \ln x$

while Leibniz insisted that the integration was only valid for positive $x$.

Leonhard Paul Euler noticed that the integration in question required an arbitrary constant, and so:

$\map \ln {-x} = \map \ln x + c$

where $c$ was necessarily imaginary.

This resolved the matter.