Elementary Complex Functions

In this lesson you’ll extend the exponential, logarithm, trigonometric, hyperbolic and power functions to $\mathbb{C}$, see which familiar properties survive and which break, and learn to handle multi-valuedness with branches and cuts.

Define

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

This is forced, not chosen: it is the only definition that is analytic and agrees with $e^x$ on the real axis, by the identity theorem.

Properties that survive:

• $e^{z_1+z_2} = e^{z_1}e^{z_2}$
• $\frac{d}{dz}e^z = e^z$
• $e^z \ne 0$ for all $z$, since $|e^z| = e^x > 0$
• $e^z$ is entire

The property that breaks, and it is a big one:

$$e^{z+2\pi i} = e^z$$

The complex exponential is periodic, with period $2\pi i$. On $\mathbb{R}$ the exponential is injective; on $\mathbb{C}$ it is infinitely many-to-one. Every consequence in this lesson traces back to that.

Geometrically $|e^z| = e^{\operatorname{Re}z}$ and $\arg e^z = \operatorname{Im}z$, so horizontal lines map to rays and vertical lines to circles. A horizontal strip of height $2\pi$ covers $\mathbb{C}\setminus\{0\}$ exactly once.

We want $\log z$ to invert $e^z$. Since the exponential is not injective, there is no single answer:

$$\log z = \ln|z| + i\arg z = \ln|z| + i(\operatorname{Arg}z + 2\pi k), \quad k\in\mathbb{Z}$$

The complex logarithm is genuinely multi-valued, taking infinitely many values differing by $2\pi i$. This is not a defect to be patched but a real feature, and it is the reason the subject has branch cuts, Riemann surfaces, and - much later - the residue theorem’s factor of $2\pi i$.

The principal logarithm picks $\operatorname{Arg}z\in(-\pi,\pi]$:

$$\operatorname{Log} z = \ln|z| + i\operatorname{Arg}z$$

To make it continuous we must delete a ray from 0 to $\infty$. The conventional choice is the negative real axis, called the principal branch cut, and on $\mathbb{C}$ minus that cut, $\operatorname{Log}$ is analytic with

$$\frac{d}{dz}\operatorname{Log}z = \frac1z$$

Two points of care that cause most of the errors in this material:

• $\log(z_1z_2) = \log z_1 + \log z_2$ holds as sets of values, but can fail for principal values. For instance with $z_1=z_2=-1$, $\operatorname{Log}(z_1z_2)=\operatorname{Log}(1)=0$, while $\operatorname{Log}(-1)+\operatorname{Log}(-1)=2\pi i$.
• $\log$ of a negative number now exists. $\operatorname{Log}(-1) = i\pi$, which is the honest answer to a question you were told had none.

Rearranging Euler’s formula gives the definitions:

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

Both are entire, both have the expected derivatives, and every algebraic identity - $\sin^2 z+\cos^2 z=1$, the addition formulas, the double-angle formulas - carries over unchanged, because the identity theorem says an identity valid on $\mathbb{R}$ must persist.

The property that breaks is boundedness:

$$\sin(iy) = i\sinh y, \qquad \cos(iy) = \cosh y$$

both of which grow without bound. Complex sine and cosine are unbounded, and they must be, by Liouville’s theorem: a bounded entire function is constant. The familiar bound $|\sin x|\le1$ is a fact about the real axis only.

Zeros are unchanged, though: $\sin z = 0$ exactly at $z = n\pi$, and $\cos z = 0$ exactly at $z = \frac{\pi}{2}+n\pi$, all real. Extending to $\mathbb{C}$ adds no new zeros.

The hyperbolic functions are the same objects rotated:

$$\sinh z = \frac{e^z-e^{-z}}{2}, \qquad \sin(iz) = i\sinh z, \qquad \cos(iz)=\cosh z$$

In $\mathbb{C}$ the distinction between trigonometric and hyperbolic functions essentially disappears, which retrospectively explains why their identities look so similar.

For general complex $a$ and $z\ne0$, define

$$z^a = e^{a\log z}$$

Because $\log$ is multi-valued, so is $z^a$ in general. The number of distinct values depends on $a$:

• $a$ an integer: one value. Nothing new.
• $a = p/q$ in lowest terms: exactly $q$ values, the $q$th roots we already know.
• $a$ irrational or complex: infinitely many values.

The famous example is

$$i^i = e^{i\log i} = e^{i(i\pi/2 + 2\pi ik)} = e^{-\pi/2 - 2\pi k}$$

Every value is real, and the principal one is $e^{-\pi/2}\approx0.2079$. That an imaginary number to an imaginary power is real is a good illustration of how much the exponential definition buys.

Also worth flagging: $(z^a)^b = z^{ab}$ can fail with principal branches, for the same reason as the logarithm identity. When in doubt, go back to $e^{a\log z}$ and track the branch.

These are logarithms in disguise. For instance

$$\arcsin z = -i\log\left(iz+\sqrt{1-z^2}\right)$$

so they are multi-valued too, and need cuts. This is the cleanest explanation of why real inverse trig functions have such awkward restricted domains: those restrictions are branch choices.

Example 1: Compute $e^{1+i\pi/2}$.

Solution.

$$e^{1+i\pi/2} = e^1\left(\cos\tfrac\pi2 + i\sin\tfrac\pi2\right) = e(0+i) = ei$$

So the answer is $ei \approx 2.718i$. Note $|e^z| = e^1 = e$ and $\arg = \pi/2$, matching. ✓

Example 2: Find all solutions of $e^z = 1$.

Solution. We need $|e^z| = e^x = 1$, so $x=0$, and $\arg e^z = y \equiv 0 \pmod{2\pi}$.

$$z = 2\pi i k, \quad k \in \mathbb{Z}$$

∎

Infinitely many solutions, where the real equation $e^x=1$ had one. That is periodicity showing up in the simplest possible way.

Example 3: Compute all values of $\log(-1)$ and the principal one.

Solution. $|-1| = 1$ so $\ln|z| = 0$, and $\arg(-1) = \pi + 2\pi k$.

$$\log(-1) = i(\pi+2\pi k), \quad k\in\mathbb{Z}$$

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

Check: $e^{i\pi} = -1$ ✓, and also $e^{3i\pi} = -1$ ✓. Both are correct logarithms, which is exactly the point.

Example 4: Compute $\operatorname{Log}(1+i)$.

Solution. $|1+i| = \sqrt2$ and $\operatorname{Arg}(1+i) = \pi/4$, so

$$\operatorname{Log}(1+i) = \ln\sqrt2 + i\frac\pi4 = \frac{\ln2}{2}+\frac{i\pi}{4} \approx 0.3466+0.7854i$$

∎

Example 5: Solve $\sin z = 2$.

Solution. No real solution exists, but a complex one does. Write $w = e^{iz}$; then $\sin z = \frac{w - w^{-1}}{2i} = 2$ becomes

$$w^2 - 4iw - 1 = 0 \quad\Longrightarrow\quad w = 2i \pm i\sqrt3 = i(2\pm\sqrt3)$$

Then $iz = \log\big(i(2\pm\sqrt3)\big)$, giving

$$z = \frac\pi2 + 2\pi k - i\ln(2\pm\sqrt3)$$

Taking $k=0$ and the plus sign, $z \approx \frac\pi2 - 1.317i$.

Check: $\sin(\frac\pi2 - 1.317i) = \cos(1.317i)\cdot 1 = \cosh(1.317) \approx 2.0$ ✓

Complex sine takes every complex value, which is Liouville’s theorem made concrete.

Example 6: Find all values of $i^i$.

Solution. $\log i = i\left(\frac\pi2+2\pi k\right)$, so

$$i^i = e^{i\log i} = e^{i\cdot i(\pi/2+2\pi k)} = e^{-\pi/2-2\pi k}$$

Every value is a positive real number, and the principal one is $e^{-\pi/2}\approx0.2079$. ∎

Example 7: Show $\log(z_1z_2) = \log z_1+\log z_2$ can fail for principal values.

Solution. Take $z_1 = z_2 = -1$. Then $z_1z_2 = 1$ and

$$\operatorname{Log}(1) = 0, \qquad \operatorname{Log}(-1)+\operatorname{Log}(-1) = i\pi+i\pi = 2\pi i$$

These differ by $2\pi i$. ∎

The rule holds as an equality of value sets, and $2\pi i$ is a member of the set $\log 1 = 2\pi i k$. The identity is true; it is the principal-value bookkeeping that fails, and that is the standard trap in this material.

AC steady state. The complex exponential $e^{i\omega t}$ replaces sines and cosines throughout electrical engineering, turning differential equations into algebra. Every phasor calculation is $e^z$ used deliberately.

Damped oscillation. A solution $e^{(-\alpha+i\omega)t}$ combines decay and oscillation in one expression, with $\operatorname{Re}$ giving the decay rate and $\operatorname{Im}$ the frequency. Reading behaviour off the position of an exponent in $\mathbb{C}$ is standard in vibration and control analysis.

Quantum phases. A wavefunction’s time evolution is multiplication by $e^{-iEt/\hbar}$, a pure phase of modulus 1. That the modulus is exactly 1 is conservation of probability.

Branch cuts in physics. In scattering theory and in Feynman propagators the choice of branch cut for a square root or logarithm encodes physical boundary conditions, and choosing wrongly gives an unphysical answer.

Digital signal processing. The $z$-transform uses complex powers $z^{-n}$, and the unit circle in the $z$-plane is the frequency axis. Evaluating a transfer function on that circle is what a frequency response is.

Logarithmic spirals. The curve $z(t)=e^{(a+ib)t}$ is a logarithmic spiral, the shape of nautilus shells, spiral galaxy arms, and hurricane bands. It is the orbit of a point under repeated multiplication by a fixed complex number.