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Complex Integration: z^2 over line segment

Complex Integration: z^2 over line segment
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2017-02-22 03:52
VectorCoordinate GeometryIntegralComplex NumberParametric Equation
In collections Mapping Diagrams to Visualize Complex Analysis
This example demonstrates the computation of the integral of z^2 along a line segment in the complex plane. Key math concepts include complex numbers, integration, and line segment parametrization. GeoGebra commands and objects like Sequence, Zip, Vector, CurveCartesian, Sphere, Intersect, Segment, and Textfield are used to construct the integration path and visualize the result, making it suitable for high school and above students learning complex integration techniques and geometric interpretations.

GeoGebra Commands & Parameters

  1. Sequence(t, t, t_0 + dt, t_1 + dt, dt) → t_3
  2. Zip(x_C(B) + (y_C(B) * ί), B, t_3) → z_1
  3. Sequence(t, t, t_0, t_1, dt) → t_*
  4. Zip(x_C(B) + (y_C(B) * ί), B, t_*) → z_*
  5. Zip(t - u, t, z_1, u, z_*) → dz_*
  6. Sequence((Element[fz_*, k] * Element[dz_*, k]), k, 1, n) → fdz_*
  7. Sequence(Sum[fdz_*, k], k, 1, n) → SUMfdz_*
  8. Zip((x(B), y(B), ht), B, fz_*) → fpoint_*
  9. Vector(z_#, f_{(z#).}) → w
  10. CurveCartesian(t, 0, ht, t, -5, 5) → Xh
  11. CurveCartesian(0, t, ht, t, -5, 5) → Yh
  12. Sphere((0, 0, 4), 1) → RS
  13. Intersect(zAxis, RS, 2) → A
  14. Segment(A, f_{(z#).}) → f_s
  15. Intersect(f_s, RS) → P_∞, C
  16. Zip((x(Z), y(Z), ht), Z, SUMfdz_*) → ESum
  17. Sequence(Element[dz_*, k], k, 1, n) → dz_n
  18. CurveCartesian(x_C(t), y_C(t), t, t_0, t_1) → c
  19. Textfield(x_C) → inputBox4
  20. Textfield(y_C) → inputBox5
  21. Vector(z_#, Pointfz_#) → u
  22. Textfield(F) → InputBox1
  23. Zip(Vector[A, B], A, z_*, B, ESum) → ESumV_*
  24. Zip(Vector[A, B], A, z_*, B, fpoint_*) → fvec_*