Riemann Surface: log(z)

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2018-03-15 18:39

Function Complex Number

This example demonstrates the Riemann surface of the logarithmic function log(z). It involves complex function concepts and is constructed using the Surface command in GeoGebra. Suitable for university-level complex function studies.

GeoGebra Commands & Parameters

Surface((u * cos(v)), (u * sin(v)), Im((u * cos(v)), (u * sin(v))), u, 0, 3, v, (-π), π - π / 200) → RSIm
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t / 2)) + (Re((u * cos(v)), (u * sin(v))) * t / 2), u, 0, 2, v, (-π), π - π / 200) → RSRe
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t)) + (Im((u * cos(v)), (u * sin(v))) * If[0 ≤ t ≤ 1, t, 1]) + (0 * (2 - t)) + (((2 * π) * (0 - 2)) * (t - 1)), u, 0, 2, v, (-π), π - π / 200) → B0
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t)) + (Im((u * cos(v)), (u * sin(v))) * If[0 ≤ t ≤ 1, t, 1]) + (0 * (2 - t)) + (((2 * π) * (1 - 2)) * (t - 1)), u, 0, 2, v, (-π), π - π / 200) → B1
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t)) + (Im((u * cos(v)), (u * sin(v))) * If[0 ≤ t ≤ 1, t, 1]) + (0 * (2 - t)) + (0 * (t - 1)), u, 0, 2, v, (-π), π - π / 200) → B2
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t)) + (Im((u * cos(v)), (u * sin(v))) * If[0 ≤ t ≤ 1, t, 1]) + (0 * (2 - t)) + (((2 * π) * (3 - 2)) * (t - 1)), u, 0, 2, v, (-π), π - π / 200) → B3
Surface((u * cos(v)), (u * sin(v)), (0 * (1 - t)) + (Im((u * cos(v)), (u * sin(v))) * If[0 ≤ t ≤ 1, t, 1]) + (0 * (2 - t)) + (((2 * π) * (4 - 2)) * (t - 1)), u, 0, 2, v, (-π), π - π / 200) → B4

AI Drawing Prompt Reference

Draw the Riemann surface of the complex function log(z), including the surfaces for the real and imaginary parts, as well as four boundary surfaces. Start by drawing the surface for the real part, then draw four boundary surfaces, each covering a different angle range.

The prompt above was compiled from the original drawing commands and is for reference only.