Complex: continuation around a branch point

Question

What values are obtained after continuously following square roots and logarithms around a nonzero circular path?

Mathematical background

An unwrapped path angle retains winding history. Along the circle, a continued square root is √r exp(iΘ/2), while a continued logarithm is ln r + iΘ.

Evaluating a selected single-valued branch is a separate operation that restricts the argument to an interval.

w(\Theta+2\pi)=-w(\Theta),\qquad w(\Theta+4\pi)=w(\Theta)

Sources: [M5], [M6]

w=\ln r+i\Theta,\qquad w(\Theta+2\pi)-w(\Theta)=2\pi i

Sources: [M6]

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

Square roots are expected to change sign after an odd number of full turns and return after an even number.

A logarithm gains 2πi per counterclockwise turn and loses 2πi per clockwise turn. Squaring or exponentiating the respective output should recover the input within numerical tolerance.

Parameters and assumptions

Use radii 1 and 2, both functions, both directions and 1–4 complete turns.
The initial angle is zero and the cut is the negative real axis. The continuous value mode is used.
Paths are sampled at 128 equal angular intervals per turn; no arbitrary-path classification is claimed.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Run one complete sampled-path case for each of the 32 settings through the shared complex kernel.

Save cumulative angles, complex inputs and outputs, and the relation residuals. Do not add a turn offset to an angle that is already unwrapped.

Methods reported by the executed kernel

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Software identifier: 0.1.0+0e6017e6a747.dirty.02665ba9098c
Git revision: 0e6017e6a747d3c610b956612b4125e232283001
Uncommitted worktree state: Present when the batch was generated
Source SHA-256: 02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b
Runtime: v26.5.1

Results from an actual run

This saved batch contains 32 completed calculations. The table and downloads below refer to those actual runs.

The largest recorded absolute diagnostic residual was 2.40178e-15. 32 of 32 residual-bearing cases were within their own reported numerical tolerance.

Across 32 recorded readings, Unwrapped angle Θ ranged from -25.13274 to 25.13274.

Across 32 recorded readings, Signed turns ranged from -4 to 4.

Across 32 recorded readings, Output Re w ranged from -1.414214 to 1.414214.

Across 32 recorded readings, Output Im w ranged from -25.13274 to 25.13274.

On the saved unit-circle square-root runs, the continued endpoint after one positive turn was (-1, 1.22465e-16); after two it was (1, -2.44929e-16). Coordinates are (Re w, Im w); small nonzero imaginary parts reflect floating-point arithmetic.

For the unit-circle logarithm, one positive turn gave Im w = 6.283185, and one negative turn gave Im w = -6.283185. The relation residual checks exponentiation at every saved path sample.

Download summary CSV Download complete summary JSON

Actual case outputs. JSON retains every observable and the complete computed data.
Case	Selected measurements	Residual and tolerance	Data
1. Square root · r=1 · 1 turns	Signed turns: 1 Output Re w: -1 Output Im w: 1.22465e-16 Maximum relation residual: 4.05742e-16	4.05742e-16 Tolerance: 2.00000e-12	JSON · CSV
2. Square root · r=1 · 2 turns	Signed turns: 2 Output Re w: 1 Output Im w: -2.44929e-16 Maximum relation residual: 6.50671e-16	6.50671e-16 Tolerance: 2.00000e-12	JSON · CSV
3. Square root · r=1 · 3 turns	Signed turns: 3 Output Re w: -1 Output Im w: 3.67394e-16 Maximum relation residual: 8.67112e-16	8.67112e-16 Tolerance: 2.00000e-12	JSON · CSV
4. Square root · r=1 · 4 turns	Signed turns: 4 Output Re w: 1 Output Im w: -4.89859e-16 Maximum relation residual: 1.11022e-15	1.11022e-15 Tolerance: 2.00000e-12	JSON · CSV
5. Square root · r=1 · -1 turns	Signed turns: -1 Output Re w: -1 Output Im w: -1.22465e-16 Maximum relation residual: 4.05742e-16	4.05742e-16 Tolerance: 2.00000e-12	JSON · CSV
6. Square root · r=1 · -2 turns	Signed turns: -2 Output Re w: 1 Output Im w: 2.44929e-16 Maximum relation residual: 6.50671e-16	6.50671e-16 Tolerance: 2.00000e-12	JSON · CSV
7. Square root · r=1 · -3 turns	Signed turns: -3 Output Re w: -1 Output Im w: -3.67394e-16 Maximum relation residual: 8.67112e-16	8.67112e-16 Tolerance: 2.00000e-12	JSON · CSV
8. Square root · r=1 · -4 turns	Signed turns: -4 Output Re w: 1 Output Im w: 4.89859e-16 Maximum relation residual: 1.11022e-15	1.11022e-15 Tolerance: 2.00000e-12	JSON · CSV
9. Logarithm · r=1 · 1 turns	Signed turns: 1 Output Re w: 0 Output Im w: 6.283185 Maximum relation residual: 3.14018e-16	3.14018e-16 Tolerance: 2.00000e-12	JSON · CSV
10. Logarithm · r=1 · 2 turns	Signed turns: 2 Output Re w: 0 Output Im w: 12.56637 Maximum relation residual: 5.55112e-16	5.55112e-16 Tolerance: 2.00000e-12	JSON · CSV
11. Logarithm · r=1 · 3 turns	Signed turns: 3 Output Re w: 0 Output Im w: 18.84956 Maximum relation residual: 8.00593e-16	8.00593e-16 Tolerance: 2.00000e-12	JSON · CSV
12. Logarithm · r=1 · 4 turns	Signed turns: 4 Output Re w: 0 Output Im w: 25.13274 Maximum relation residual: 1.04738e-15	1.04738e-15 Tolerance: 2.00000e-12	JSON · CSV
13. Logarithm · r=1 · -1 turns	Signed turns: -1 Output Re w: 0 Output Im w: -6.283185 Maximum relation residual: 3.14018e-16	3.14018e-16 Tolerance: 2.00000e-12	JSON · CSV
14. Logarithm · r=1 · -2 turns	Signed turns: -2 Output Re w: 0 Output Im w: -12.56637 Maximum relation residual: 5.55112e-16	5.55112e-16 Tolerance: 2.00000e-12	JSON · CSV
15. Logarithm · r=1 · -3 turns	Signed turns: -3 Output Re w: 0 Output Im w: -18.84956 Maximum relation residual: 8.00593e-16	8.00593e-16 Tolerance: 2.00000e-12	JSON · CSV
16. Logarithm · r=1 · -4 turns	Signed turns: -4 Output Re w: 0 Output Im w: -25.13274 Maximum relation residual: 1.04738e-15	1.04738e-15 Tolerance: 2.00000e-12	JSON · CSV
17. Square root · r=2 · 1 turns	Signed turns: 1 Output Re w: -1.414214 Output Im w: 1.73191e-16 Maximum relation residual: 1.13221e-15	1.13221e-15 Tolerance: 3.00000e-12	JSON · CSV
18. Square root · r=2 · 2 turns	Signed turns: 2 Output Re w: 1.414214 Output Im w: -3.46382e-16 Maximum relation residual: 1.37503e-15	1.37503e-15 Tolerance: 3.00000e-12	JSON · CSV
19. Square root · r=2 · 3 turns	Signed turns: 3 Output Re w: -1.414214 Output Im w: 5.19574e-16 Maximum relation residual: 1.91010e-15	1.91010e-15 Tolerance: 3.00000e-12	JSON · CSV
20. Square root · r=2 · 4 turns	Signed turns: 4 Output Re w: 1.414214 Output Im w: -6.92765e-16 Maximum relation residual: 2.40178e-15	2.40178e-15 Tolerance: 3.00000e-12	JSON · CSV
21. Square root · r=2 · -1 turns	Signed turns: -1 Output Re w: -1.414214 Output Im w: -1.73191e-16 Maximum relation residual: 1.13221e-15	1.13221e-15 Tolerance: 3.00000e-12	JSON · CSV
22. Square root · r=2 · -2 turns	Signed turns: -2 Output Re w: 1.414214 Output Im w: 3.46382e-16 Maximum relation residual: 1.37503e-15	1.37503e-15 Tolerance: 3.00000e-12	JSON · CSV
23. Square root · r=2 · -3 turns	Signed turns: -3 Output Re w: -1.414214 Output Im w: -5.19574e-16 Maximum relation residual: 1.91010e-15	1.91010e-15 Tolerance: 3.00000e-12	JSON · CSV
24. Square root · r=2 · -4 turns	Signed turns: -4 Output Re w: 1.414214 Output Im w: 6.92765e-16 Maximum relation residual: 2.40178e-15	2.40178e-15 Tolerance: 3.00000e-12	JSON · CSV
25. Logarithm · r=2 · 1 turns	Signed turns: 1 Output Re w: 0.6931472 Output Im w: 6.283185 Maximum relation residual: 6.28037e-16	6.28037e-16 Tolerance: 3.00000e-12	JSON · CSV
26. Logarithm · r=2 · 2 turns	Signed turns: 2 Output Re w: 0.6931472 Output Im w: 12.56637 Maximum relation residual: 1.11022e-15	1.11022e-15 Tolerance: 3.00000e-12	JSON · CSV
27. Logarithm · r=2 · 3 turns	Signed turns: 3 Output Re w: 0.6931472 Output Im w: 18.84956 Maximum relation residual: 1.60119e-15	1.60119e-15 Tolerance: 3.00000e-12	JSON · CSV
28. Logarithm · r=2 · 4 turns	Signed turns: 4 Output Re w: 0.6931472 Output Im w: 25.13274 Maximum relation residual: 2.09476e-15	2.09476e-15 Tolerance: 3.00000e-12	JSON · CSV
29. Logarithm · r=2 · -1 turns	Signed turns: -1 Output Re w: 0.6931472 Output Im w: -6.283185 Maximum relation residual: 6.28037e-16	6.28037e-16 Tolerance: 3.00000e-12	JSON · CSV
30. Logarithm · r=2 · -2 turns	Signed turns: -2 Output Re w: 0.6931472 Output Im w: -12.56637 Maximum relation residual: 1.11022e-15	1.11022e-15 Tolerance: 3.00000e-12	JSON · CSV
31. Logarithm · r=2 · -3 turns	Signed turns: -3 Output Re w: 0.6931472 Output Im w: -18.84956 Maximum relation residual: 1.60119e-15	1.60119e-15 Tolerance: 3.00000e-12	JSON · CSV
32. Logarithm · r=2 · -4 turns	Signed turns: -4 Output Re w: 0.6931472 Output Im w: -25.13274 Maximum relation residual: 2.09476e-15	2.09476e-15 Tolerance: 3.00000e-12	JSON · CSV

Interpretation

Compare the signs or imaginary increments in the saved endpoint values while checking the defining relation along the path.

A coincident input point does not imply the same continued output. Changing a branch cut would choose another single-valued representation without changing that relation.

Limitations and possible errors

The experiment follows controlled circles, not arbitrary loops.
The three-dimensional view is a projection of a complex relation and can overlap. Zero is outside the sampled logarithm domain.

The kernel reported no warnings for this batch. This is a recorded software status, not an independent review.

Reproduce this study

Restore the function, radius, signed cumulative angle and continuous-value mode in Complex.
Use JSON to preserve the complete sampled path and the initial-branch convention; use CSV for pointwise relation checks.

Open complex

1. Square root · r=1 · 1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": 6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 6.2831853 rad
Signed turns: 1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -2.449e-16
Output Re w: -1
Output Im w: 1.225e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1
Continued Im w: 1.225e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 4.057e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

2. Square root · r=1 · 2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": 12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 12.566371 rad
Signed turns: 2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -4.899e-16
Output Re w: 1
Output Im w: -2.449e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1
Continued Im w: -2.449e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 6.507e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

3. Square root · r=1 · 3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": 18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 18.849556 rad
Signed turns: 3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -7.348e-16
Output Re w: -1
Output Im w: 3.674e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1
Continued Im w: 3.674e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 8.671e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

4. Square root · r=1 · 4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": 25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 25.132741 rad
Signed turns: 4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -9.797e-16
Output Re w: 1
Output Im w: -4.899e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1
Continued Im w: -4.899e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 1.110e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

5. Square root · r=1 · -1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": -6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -6.2831853 rad
Signed turns: -1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 2.449e-16
Output Re w: -1
Output Im w: -1.225e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1
Continued Im w: -1.225e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 4.057e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

6. Square root · r=1 · -2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": -12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -12.566371 rad
Signed turns: -2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 4.899e-16
Output Re w: 1
Output Im w: 2.449e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1
Continued Im w: 2.449e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 6.507e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

7. Square root · r=1 · -3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": -18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -18.849556 rad
Signed turns: -3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 7.348e-16
Output Re w: -1
Output Im w: -3.674e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1
Continued Im w: -3.674e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 8.671e-16Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

8. Square root · r=1 · -4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "sqrt",
  "radius": 1,
  "angle": -25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -25.132741 rad
Signed turns: -4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 9.797e-16
Output Re w: 1
Output Im w: 4.899e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1
Continued Im w: 4.899e-16
Principal value Re w: 1
Principal value Im w: 0
Maximum relation residual: 1.110e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

9. Logarithm · r=1 · 1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": 6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 6.2831853 rad
Signed turns: 1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -2.449e-16
Output Re w: 0
Output Im w: 6.2831853
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: 6.2831853
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 3.140e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

10. Logarithm · r=1 · 2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": 12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 12.566371 rad
Signed turns: 2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -4.899e-16
Output Re w: 0
Output Im w: 12.566371
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: 12.566371
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 5.551e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

11. Logarithm · r=1 · 3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": 18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 18.849556 rad
Signed turns: 3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -7.348e-16
Output Re w: 0
Output Im w: 18.849556
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: 18.849556
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 8.006e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

12. Logarithm · r=1 · 4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": 25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 25.132741 rad
Signed turns: 4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: -9.797e-16
Output Re w: 0
Output Im w: 25.132741
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: 25.132741
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 1.047e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

13. Logarithm · r=1 · -1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": -6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -6.2831853 rad
Signed turns: -1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 2.449e-16
Output Re w: 0
Output Im w: -6.2831853
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: -6.2831853
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 3.140e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

14. Logarithm · r=1 · -2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": -12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -12.566371 rad
Signed turns: -2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 4.899e-16
Output Re w: 0
Output Im w: -12.566371
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: -12.566371
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 5.551e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

Download complete case JSON · Download case CSV

15. Logarithm · r=1 · -3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": -18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -18.849556 rad
Signed turns: -3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 7.348e-16
Output Re w: 0
Output Im w: -18.849556
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: -18.849556
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 8.006e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

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16. Logarithm · r=1 · -4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "log",
  "radius": 1,
  "angle": -25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -25.132741 rad
Signed turns: -4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 1
Input Im z: 9.797e-16
Output Re w: 0
Output Im w: -25.132741
Value convention: Continuous from Θ = 0
Continued Re w: 0
Continued Im w: -25.132741
Principal value Re w: 0
Principal value Im w: 0
Maximum relation residual: 1.047e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

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17. Square root · r=2 · 1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": 6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 6.2831853 rad
Signed turns: 1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -4.899e-16
Output Re w: -1.4142136
Output Im w: 1.732e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1.4142136
Continued Im w: 1.732e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.132e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

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18. Square root · r=2 · 2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": 12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 12.566371 rad
Signed turns: 2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -9.797e-16
Output Re w: 1.4142136
Output Im w: -3.464e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1.4142136
Continued Im w: -3.464e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.375e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

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19. Square root · r=2 · 3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": 18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 18.849556 rad
Signed turns: 3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -1.470e-15
Output Re w: -1.4142136
Output Im w: 5.196e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1.4142136
Continued Im w: 5.196e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.910e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

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20. Square root · r=2 · 4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": 25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 25.132741 rad
Signed turns: 4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -1.959e-15
Output Re w: 1.4142136
Output Im w: -6.928e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1.4142136
Continued Im w: -6.928e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 2.402e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

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21. Square root · r=2 · -1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": -6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -6.2831853 rad
Signed turns: -1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 4.899e-16
Output Re w: -1.4142136
Output Im w: -1.732e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1.4142136
Continued Im w: -1.732e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.132e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

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22. Square root · r=2 · -2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": -12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -12.566371 rad
Signed turns: -2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 9.797e-16
Output Re w: 1.4142136
Output Im w: 3.464e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1.4142136
Continued Im w: 3.464e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.375e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

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23. Square root · r=2 · -3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": -18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -18.849556 rad
Signed turns: -3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 1.470e-15
Output Re w: -1.4142136
Output Im w: -5.196e-16
Value convention: Continuous from Θ = 0
Continued Re w: -1.4142136
Continued Im w: -5.196e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 1.910e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

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24. Square root · r=2 · -4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "sqrt",
  "radius": 2,
  "angle": -25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -25.132741 rad
Signed turns: -4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 1.959e-15
Output Re w: 1.4142136
Output Im w: 6.928e-16
Value convention: Continuous from Θ = 0
Continued Re w: 1.4142136
Continued Im w: 6.928e-16
Principal value Re w: 1.4142136
Principal value Im w: 0
Maximum relation residual: 2.402e-15Maximum |w² − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

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25. Logarithm · r=2 · 1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": 6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 6.2831853 rad
Signed turns: 1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -4.899e-16
Output Re w: 0.69314718
Output Im w: 6.2831853
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: 6.2831853
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 6.280e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

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26. Logarithm · r=2 · 2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": 12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 12.566371 rad
Signed turns: 2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -9.797e-16
Output Re w: 0.69314718
Output Im w: 12.566371
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: 12.566371
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 1.110e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

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27. Logarithm · r=2 · 3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": 18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 18.849556 rad
Signed turns: 3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -1.470e-15
Output Re w: 0.69314718
Output Im w: 18.849556
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: 18.849556
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 1.601e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

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28. Logarithm · r=2 · 4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": 25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: 25.132741 rad
Signed turns: 4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: -1.959e-15
Output Re w: 0.69314718
Output Im w: 25.132741
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: 25.132741
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 2.095e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

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29. Logarithm · r=2 · -1 turns

Recorded 17 September 2026 at 00:39 UTC. 129 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": -6.283185307179586,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -6.2831853 rad
Signed turns: -1Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 4.899e-16
Output Re w: 0.69314718
Output Im w: -6.2831853
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: -6.2831853
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 6.280e-16Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 129128 angular intervals per complete turn, including the start and end.

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30. Logarithm · r=2 · -2 turns

Recorded 17 September 2026 at 00:39 UTC. 257 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": -12.566370614359172,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -12.566371 rad
Signed turns: -2Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 9.797e-16
Output Re w: 0.69314718
Output Im w: -12.566371
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: -12.566371
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 1.110e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 257128 angular intervals per complete turn, including the start and end.

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31. Logarithm · r=2 · -3 turns

Recorded 17 September 2026 at 00:39 UTC. 385 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": -18.84955592153876,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -18.849556 rad
Signed turns: -3Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 1.470e-15
Output Re w: 0.69314718
Output Im w: -18.849556
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: -18.849556
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 1.601e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 385128 angular intervals per complete turn, including the start and end.

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32. Logarithm · r=2 · -4 turns

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "fn": "log",
  "radius": 2,
  "angle": -25.132741228718345,
  "cutAngle": 3.141592653589793,
  "valueMode": "continuous",
  "projection": "real"
}

Model version: complex-1.0.0

Float64 polar evaluation; unwrapped circular path; separate half-open branch argument; 128 equal angular intervals per turn; direct defining-relation checks.

Unwrapped angle Θ: -25.132741 rad
Signed turns: -4Θ / 2π for this controlled circle; fractional values describe an open path.
Input Re z: 2
Input Im z: 1.959e-15
Output Re w: 0.69314718
Output Im w: -25.132741
Value convention: Continuous from Θ = 0
Continued Re w: 0.69314718
Continued Im w: -25.132741
Principal value Re w: 0.69314718
Principal value Im w: 0
Maximum relation residual: 2.095e-15Maximum |exp(w) − z| across both value conventions and every saved path sample.
Path samples: 513128 angular intervals per complete turn, including the start and end.

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Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

[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.