Complex Logarithm

Definition

The complex logarithm is the inverse of the complex exponential. Because $e^z$ is periodic, the logarithm is a multi-valued function: $\log z = \ln |z| + i(\arg z + 2k\pi)$

• How to read: “The log of z equals the natural log of the absolute value of z plus i times the quantity arg z plus two k pi.”
• Meaning: Every non-zero $z$ has infinitely many logarithms differing by integer multiples of $2\pi i$ in the imaginary part.

Why It Matters

It reveals the multi-valued nature of rotations and growth, allowing for the precise calculation of phase and magnitude in complex systems.

Core Concepts

• Principal Value ($\text{Log } z$): Defined using the principal argument $\text{Arg } z \in (-\pi, \pi]$.
  • How to read: “The log of z equals the natural log of the absolute value of z plus i times the principal argument z, where the principal argument z is between negative pi and pi.”
  • Meaning: Principal (single-valued) branch—picks one representative angle per $z$ using the principal argument.
• Branch Cuts: To make $\text{Log } z$ a continuous, single-valued function, we must remove a ray from the plane (usually the negative real axis). This is called a branch cut.
• Complex Powers: $z^c$ is defined as $e^{c\log z}$. This implies that $z^c$ is also multi-valued (e.g., $i^i$ has infinitely many real values).
  • How to read: “The value z to the c equals e to the power c times log z.”
  • Meaning: Powers inherit the multi-valuedness of the logarithm—different branches give different values.
• Discontinuity: Crossing a branch cut results in a “jump” of $2\pi i$ in the function value.

Connected Concepts

• Complex Numbers: Operations
• Polar Form of Complex Numbers