Complex Logarithm Branches

For any nonzero complex number z = r · exp(iθ) with positive modulus r > 0, the complex logarithm is multivalued because angles differing by integer multiples of 2π map to the same point in the complex plane. Separating the single-valued real part ln(r) from the phase yields three distinct concepts:

Interactive Experiment

Inspect branch jumps and continuous winding in the Complex Logarithm Branches interactive workspace:

1. Set Radius r = 2, Angle theta over pi = 1/2, and Branch k = 0. Verify Point real = 0, Point imaginary = 2, Log real part = 0.6931471805599453, and zero continuation shift.
2. Set Radius r = 1, Angle theta over pi = 3/2, and Branch k = -1. Observe that Principal argument over pi = -0.5 and Continued imaginary over pi = 1.5 differ by Continuation shift over pi = 2.
3. Set Radius r = 1, Angle theta over pi = -1, and Branch k = 0. At z = -1 on the negative real axis, the upper-side convention sets Principal argument over pi = 1 while the clockwise path has Continued imaginary over pi = -1 (shift -2).
4. Compare Angle theta over pi = 2 with Branch k = 0 (one full positive winding returning to z = 1 with shift 2) against Angle theta over pi = 0 with Branch k = -1 (selected branch -2, shift 0).