Complex logarithm: log(z) = ln|z| + i·arg(z). Multi-valued! Principal value: Log(z) = ln|z| + i·Arg(z), where Arg ∈ (−π, π]. Branch cut along negative real axis. General: log(z) = Log(z) + 2kπi for k ∈ Z.
How to Use
Enter a complex number:
- z = x + iy: Input
- Log(z): Principal value
- Branches: k = -2,...,2
Branch Cuts
log(z) is multi-valued because e^w is periodic (period 2πi). To make single-valued, cut the plane. Principal: cut along (−∞, 0]. Other choices: cut along any ray from 0. The cut makes log analytic on the slit plane.
Properties
- Log(z₁z₂) = Log(z₁)+Log(z₂) (mod 2πi)
- Log(e^z) = z only if Im(z) ∈ (−π, π]
- d/dz Log(z) = 1/z (on slit plane)
Step-by-Step Instructions
- 1Enter Re(z) and Im(z).
- 2Get |z| and Arg(z).
- 3View Log(z).
- 4See other branches.
- 5Check branch cut.