Elementary Analytic Functions

Elementary functions extend naturally from real to complex analysis, but gain new properties such as periodicity, multivaluedness, and branch cuts. These functions are fundamental examples of holomorphic and meromorphic functions.

The Exponential Function

Definition2.5Complex exponential

The complex exponential is defined by

$e^z = e^{x+iy} = e^x(\cos y + i\sin y)$

where $z = x + iy$ . Equivalently, $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$ (power series with infinite radius of convergence).

RemarkProperties of exponential

$e^{z+w} = e^z e^w$ for all $z, w \in \mathbb{C}$ .
$|e^z| = e^{\text{Re}(z)}$ and $\arg(e^z) = \text{Im}(z)$ (mod $2\pi$ ).
$e^z$ is periodic with period $2\pi i$ : $e^{z + 2\pi i} = e^z$ .
$e^z$ is entire and never zero.
$(e^z)' = e^z$ .

ExampleExponential of pure imaginary

For $z = iy$ where $y \in \mathbb{R}$ :

$e^{iy} = \cos y + i\sin y.$

This is Euler's formula. In particular, $e^{i\pi} = -1$ and $e^{2\pi i} = 1$ .

Trigonometric Functions

Definition2.6Complex sine and cosine

$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}.$

RemarkProperties

$\sin^2 z + \cos^2 z = 1$ (Pythagorean identity holds).
$\sin z$ and $\cos z$ are entire.
$(\sin z)' = \cos z$ and $(\cos z)' = -\sin z$ .
$\sin z$ and $\cos z$ are periodic with period $2\pi$ .
Unlike real sine and cosine, they are unbounded: $|\sin(iy)| = \sinh y \to \infty$ as $y \to \infty$ .

ExampleSine is unbounded

$\sin(iy) = \frac{e^{-y} - e^y}{2i} = i\sinh y$ . For large real $y$ , $|\sin(iy)| = \sinh y \approx \frac{e^y}{2} \to \infty$ . Thus $\sin z$ is unbounded on $\mathbb{C}$ , unlike the real sine function.

Hyperbolic Functions

Definition2.7Hyperbolic functions

$\sinh z = \frac{e^z - e^{-z}}{2}, \quad \cosh z = \frac{e^z + e^{-z}}{2}.$

RemarkRelation to trigonometric functions

$\sin(iz) = i\sinh z$
$\cos(iz) = \cosh z$
$\sinh(iz) = i\sin z$
$\cosh(iz) = \cos z$

The Complex Logarithm

Definition2.8Complex logarithm (multivalued)

The complex logarithm is the inverse of the exponential function. For $z = re^{i\theta}$ with $r > 0$ :

$\log z = \ln r + i(\theta + 2\pi k), \quad k \in \mathbb{Z}.$

The logarithm is multivalued: infinitely many values differ by integer multiples of $2\pi i$ .

RemarkPrincipal branch

To make $\log z$ single-valued, we choose a branch: a connected region where $\arg(z)$ varies continuously. The principal branch is defined by restricting $\theta \in (-\pi, \pi]$ :

$\text{Log } z = \ln |z| + i\arg(z), \quad -\pi < \arg(z) \leq \pi.$

The branch cut is the negative real axis $\mathbb{R}_{\leq 0}$ . On $\mathbb{C} \setminus \mathbb{R}_{\leq 0}$ , $\text{Log } z$ is holomorphic with derivative $(\text{Log } z)' = 1/z$ .

ExampleMultiple values

For $z = -1 = e^{i\pi}$ :

$\log(-1) = i\pi, \; i3\pi, \; i5\pi, \; \ldots, \; -i\pi, \; -i3\pi, \; \ldots$

The principal value is $\text{Log}(-1) = i\pi$ .

Complex Powers

Definition2.9Complex power

For $z, w \in \mathbb{C}$ with $z \neq 0$ :

$z^w = e^{w \log z}.$

This is generally multivalued (unless $w$ is an integer).

ExampleMultivalued power

$i^i = e^{i \log i} = e^{i(i\pi/2 + 2\pi i k)} = e^{-\pi/2 - 2\pi k}$ for $k \in \mathbb{Z}$ . The principal value is $i^i = e^{-\pi/2} \approx 0.2079$ , which is real!

ExampleSquare root

$\sqrt{z} = z^{1/2} = e^{(1/2)\log z}$ . For instance, $\sqrt{i} = e^{(1/2)(i\pi/2 + 2\pi i k)} = e^{i\pi/4 + \pi i k}$ .

The two values are $e^{i\pi/4} = \frac{1}{\sqrt{2}}(1+i)$ and $e^{i5\pi/4} = -\frac{1}{\sqrt{2}}(1+i)$ .

Inverse Trigonometric Functions

RemarkInverse trig functions

$\arcsin z$ , $\arccos z$ , $\arctan z$ are defined as inverses of $\sin z$ , $\cos z$ , $\tan z$ . They are multivalued and can be expressed in terms of the complex logarithm:

$\arcsin z = -i \log(iz + \sqrt{1 - z^2}), \quad \arccos z = -i \log(z + i\sqrt{1-z^2}).$

These functions have branch cuts on the real axis outside $[-1, 1]$ (for $\arcsin$ and $\arccos$ ).

Summary

RemarkKey features of elementary functions

Exponential: entire, periodic, never zero.
Trigonometric: entire, periodic, unbounded.
Logarithm: multivalued, requires branch cuts, holomorphic on domains avoiding cuts.
Powers: multivalued (unless exponent is an integer).
Inverse trig: multivalued, branch cuts on real axis.

These functions are the building blocks for all of complex analysis, appearing in power series, residue calculations, and conformal maps.