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Mapping Diagram for z^n with Curve on Riemann Surface

Mapping Diagram for z^n with Curve on Riemann Surface
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2016-04-04 04:37
FunctionComplex NumberParametric EquationGeometric ConstructionGraphing
In collections Mapping Diagrams to Visualize Complex Analysis
This example demonstrates the mapping diagram for the complex power function z^n, with a visualization of the image curve on the Riemann surface. Key math concepts include complex numbers, mapping, Riemann surfaces, and parametric equations. Construction highlights involve GeoGebra commands like Sequence, CurveCartesian, Textfield, Vector, Locus, Intersect, Segment, and Angle, along with objects such as t_#, r, e, z_*, n, b, a, height_{image}, C_{image}, fz_*, C_{domain}, γ_{(t)}, f_{γ(t)}, f_{(z#).}, fpoint_*, fvec_*, d, inputBox1, w, inputBox2, Xh, Yh, inputBox3, c, text2, loc1, text3, A, δ, B, t, u, f, text4, text5, text6, text7, text8, wndg_{#}, text1. Suitable for high school and above students to understand the geometric meaning of complex functions and the construction of Riemann surfaces.

GeoGebra Commands & Parameters

  1. Sequence((r * (cos(t) + (sin(t) * ί))), t, -3.141592653589793, 3.141592653589793, 0.5235987755982988) → z_*
  2. Sequence((x(Element[fz_*, k]), y(Element[fz_*, k]), height_{image}), k, 1, 12) → fpoint_*
  3. Sequence(Vector[Element[z_*, j], Element[fpoint_*, j]], j, 1, 12) → fvec_*
  4. CurveCartesian((r * cos(t)), (r * sin(t)), height_{image}, t, 0, 6.283185307179586) → d
  5. Textfield(fz_*) → inputBox1
  6. Vector(γ_{(t)}, f_{(z#).}) → w
  7. Textfield(f_{γ(t)}) → inputBox2
  8. CurveCartesian(t, 0, height_{image}, t, -5, 5) → Xh
  9. CurveCartesian(0, t, height_{image}, t, -5, 5) → Yh
  10. Textfield(z_*) → inputBox3
  11. Locus(f_{(z#).}, t_#) → loc1
  12. Intersect(xAxis, yAxis) → A
  13. Segment(γ_{(t)}, A) → δ
  14. Intersect(c, xAxis, 2) → B
  15. Angle(B, A, γ_{(t)}) → t
  16. Vector(f_{γ(t)}, f_{(z#).}) → u
  17. Sequence((1, 0, t / 4 + 3), t, 0, (4 * π), (2 * π / n)) → wndg_{#}