Complex logarithm

The complex logarithm is the inverse of the Complex exponential: $\log z=w$ means $e^w=z$.

Because $e^z$ is periodic with period $2\pi i$, the inverse is multi-valued. For each $z\ne 0$, there are infinitely many $w$ satisfying $e^w=z$, differing from each other by integer multiples of $2\pi i$.

This multi-valuedness is the genuinely new phenomenon of complex analysis: it has no analog in real-variable calculus, where $\log x$ for $x>0$ is a perfectly nice single-valued function.

Derivation

Write $z=r{e}^{i\theta }$ with $r>0$ and $\theta$ any argument of $z$. Then $w=\ln r+i\theta$ satisfies

$e^w=e^{\ln r}\cdot e^{i\theta }=r\cdot e^{i\theta }=z.$

But $\theta$ is only defined mod $2\pi$. So $w$ is not unique — any of

$w=\ln |z|+i({\theta }_0+2\pi n),n\in \mathbb{Z}$

(with ${\theta }_0$ any one choice of argument) is a valid logarithm.

Principal value

The standard single-valued choice. Take $\theta =\mathrm{Arg}(z)\in (-\pi ,\pi ]$ — the principal argument — and define

$\mathrm{Log}z=\ln |z|+i\mathrm{Arg}(z),z\ne 0.$

Capital “L” for the principal branch. On the slit plane $\mathbb{C}\setminus (-\infty ,0]$, $\mathrm{Log}$ is single-valued and analytic, with derivative

$\frac{d}{dz}\mathrm{Log}z=\frac{1}{z}.$

The negative real axis is the branch cut: across it, $\mathrm{Arg}$ jumps by $2\pi$, making $\mathrm{Log}$ discontinuous. Approach $-1$ from above ($e^{i(\pi -\epsilon )}$): $\mathrm{Log}\to i\pi$. From below ($e^{-i(\pi -\epsilon )}$): $\mathrm{Log}\to -i\pi$. Difference: $2\pi i$.

General logarithm

Not a function but a multi-valued expression:

$\log z=\mathrm{Log}z+2\pi in,n\in \mathbb{Z}.$

In specific calculations, you choose a branch (a continuous single-valued function on some smaller domain) and stick with it. Standard choices: the principal branch, or one of its $2\pi$-shifted relatives.

Worked examples

$\mathrm{Log}(-1)$. $-1$ has modulus $1$, principal argument $\pi$. So $\mathrm{Log}(-1)=\ln 1+i\pi =i\pi$.

But $\log (-1)=i\pi +2\pi in$ — the full multi-valued set is $\{\dots ,-3i\pi ,-i\pi ,i\pi ,3i\pi ,\dots \}$.

$\mathrm{Log}(i)$. $i$ has modulus $1$, argument $\pi /2$. So $\mathrm{Log}(i)=i\pi /2$.

$\mathrm{Log}(-1+i)$. Modulus $\sqrt{2}$, principal argument $3\pi /4$ (second quadrant). So $\mathrm{Log}(-1+i)=\frac{1}{2}\ln 2+i(3\pi /4)$.

Identity caveats

The familiar identity $\log (z_1{z}_2)=\log z_1+\log z_2$ does not always hold for the principal branch — sometimes the sum’s argument falls outside $(-\pi ,\pi ]$ and needs an adjustment by $\pm 2\pi i$.

The multi-valued identity holds modulo $2\pi i$. Be careful with branches when manipulating logarithms symbolically.

Why a branch cut

The function $\mathrm{Log}$ cannot be made continuous on all of $\mathbb{C}\setminus \{0\}$: any single-valued logarithm must jump somewhere. The cut $(-\infty ,0]$ is the conventional choice but not forced — you could equally well slit along any ray from origin to infinity, and define a different principal branch. The geometric obstruction is the non-simply-connectedness of $\mathbb{C}\setminus \{0\}$.

The complex logarithm is the simplest example of a multi-valued function in complex analysis. The full theory uses Riemann surfaces — multi-sheeted covering spaces that “unwrap” the multi-valuedness into a single-valued analytic function on an enlarged domain. Vector Calculus and Complex Analysis doesn’t develop Riemann surfaces, but the branch-cut framework handles all the practical cases.

In context

• Complex power $z^c$ is defined as $e^{c\log z}$, inheriting multi-valuedness from $\log z$.
• The integrand $1/z$ in contour integrals integrates to $\log z$ — and the multi-valuedness is what creates the $2\pi i$ contribution to $\oint dz/z$ around the origin. The Residue theorem systematizes this.
• In physics, branch cuts often represent physical discontinuities (the surface of a square root in computing transmission coefficients, or branch points in scattering amplitudes).