Reasoning
Read the definitions, follow the arguments and examine the conditions of each result.
Background: Complex arithmetic, Polar coordinates, Complex exponentials. Start with the model for a brief introduction, follow the guided exploration for a closer look, or work through the arguments and experiment for a detailed study.
What you can learn
- Track an unwrapped angle through several turns.
- Distinguish continuation from evaluation on a branch.
- Interpret a three-dimensional projection of a complex relation.
Reading step 1 of 5: Keep the path angle
Step 1 · Derivation · Selected
Keep the path angle
Write a nonzero input as r exp(iΘ). For a point alone, adding 2π to an argument does not change the input. For a followed path, however, Θ records how the point arrived there. Reducing Θ modulo 2π discards that history and changes the question the calculation can answer.
This exhibit follows controlled circular paths. Counterclockwise motion increases the unwrapped angle; clockwise motion decreases it. A fractional number of traversed turns is meaningful for an unfinished path, but an integer winding number requires a closed path. Changing the displayed branch cut does not alter the stored path angle.
Sources: [M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Step 2 · Derivation
Continue a square root
Choose the positive root when the input starts on the positive real axis. Along the circle, w = √r exp(iΘ/2) changes continuously and its square is r exp(iΘ), the input. Increasing Θ by 2π multiplies w by exp(iπ) = −1. Increasing it by 4π multiplies w by exp(2πi) = 1.
The input can therefore return to its starting point while the followed value changes. This is compatible with w² = z at every point. It also prevents a continuous single-valued square-root choice on the entire punctured plane: following such a choice around this loop would require the starting value to equal its negative.
Derivation. A continuously followed square root changes sign after one full turn and returns after two.
Conditions and assumptions
- r > 0; one starting square root is fixed and continued along the path.
- z = r exp(iΘ) with r > 0.
- Θ is the continuous, unwrapped path angle.
- The starting root is fixed.
Sources: [M5] Complex Analysis: A Visual and Interactive Introduction — Riemann Surfaces; [M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Step 3 · Derivation
Derive the logarithm values
Solve exp(u + iv) = r exp(iθ). Comparing magnitudes gives u = ln r. Comparing arguments gives v = θ + 2πk for an integer k. These are all logarithm values at the same nonzero input.
Along a continuous path, use the already unwrapped angle Θ and write w = ln r + iΘ. Any initial branch offset is chosen once. Adding a turn counter again would count the same winding twice. A counterclockwise loop adds 2πi, a clockwise loop subtracts it, and exponentiating any of these continued values recovers the input.
Derivation. A continued logarithm gains 2πi on each counterclockwise turn.
Conditions and assumptions
- A nonzero circular path.
- The initial logarithm branch is fixed once.
- The turn count is not added again to Θ.
Sources: [M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Step 4 · Derivation
Separate a branch from continuation
A single-valued branch restricts the argument to an interval between two copies of a chosen cut direction. The principal convention used here gives arguments in (−π,π]; its analytic branch excludes the negative-real-axis cut. A point value assigned on that ray by convention is not evidence of analyticity there.
Moving the cut chooses a different single-valued branch and can move the location of its jump. It leaves the underlying multi-valued relation unchanged. The workspace calculates the selected-branch value and the continuously followed value separately, so their different behavior at a crossing can be compared directly.
Sources: [M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Step 5 · Visualization
Interpret the surface projection
The relation between complex input and output has four real coordinates. The square-root display keeps the two input coordinates and one output coordinate as position; labels and color retain the other output coordinate. Overlap in this projection is possible. For the logarithm, the vertical coordinate records the accumulated angle, and finite visible turns are only a viewing limit.
The origin is a branch point of the square-root projection. The surface itself has the smooth parametrization w ↦ (w²,w), so the branching does not make that surface singular. The logarithm is undefined at zero. Exported samples check w² against z or exp(w) against z, preserving radius, direction and angle history as reproducible inputs.
Visualization. The surface display retains three real coordinates and encodes the remaining information in labels and color.
Conditions and assumptions
- The displayed projection may overlap.
- Finite displayed turns are a viewport limit, not a limit on logarithm branches.
Sources: [M5] Complex Analysis: A Visual and Interactive Introduction — Riemann Surfaces; [M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
References and review status
Project attribution: BIPH Math Society. The exposition was prepared with tool assistance.
Review status: Not reviewed. Independent review by a teacher or society member has not yet been recorded. Numerical checks do not replace mathematical review.
[M5] Complex Analysis: A Visual and Interactive Introduction — Riemann Surfaces
Juan Carlos Ponce Campuzano
Author’s visual treatment of multi-valued complex functions, including separate real-component and imaginary-component surface representations. The project builds its own sampled geometry and explains which coordinates each projection retains. Source checked 16 September 2026.
[M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Juan Carlos Ponce Campuzano
Derivation of logarithm values from exp(w)=z, principal-value conventions and analytic branches on cut domains. The exhibit distinguishes a conventionally assigned boundary value from an analytic branch domain. Source checked 16 September 2026.