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Taylor Polynomials for Complex Functions: Mapping Diagrams

Taylor Polynomials for Complex Functions: Mapping Diagrams
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2019-11-19 01:43
FunctionSequenceCalculusComplex Number
In collections Mapping Diagrams to Visualize Complex Analysis
This example illustrates Taylor polynomials for complex functions and their mapping diagrams. Key mathematical concepts include complex functions, Taylor polynomials, and mapping diagrams. It utilizes commands like Sequence, Zip, Derivative, Sum, and If in GeoGebra. Suitable for university-level mathematics education and research.

GeoGebra Commands & Parameters

  1. Sequence(0, n) → N
  2. Zip(x^(nn), nn, N) → l3
  3. Derivative(F, z) → Fz
  4. Derivative(Fz, z) → Fzz
  5. Derivative(Fzz, z) → Fzzz
  6. Derivative(Fzzz, z) → Fzzzz
  7. Sequence(0, m) → M
  8. Zip(((2 * pi) * k / m), k, M) → t_*
  9. Zip((δ * ℯ^((t * ί))) + z_#, t, t_*) → z_*
  10. Zip(Simplify[Derivative[f, nn]] / nn!, nn, N) → l2
  11. Zip((xx(z) * yy(x)), xx, l2, yy, l3) → l4
  12. Sum(l4) → q
  13. If(e ≟ 1, z_# - fz_#, 0) → align
  14. Circle(z_#, δ) → c
  15. Zip(f(z) + align, z, z_*) → fz_*
  16. Zip((x(z), y(z), ht), z, fz_*) → fpt_*
  17. Line(O, ZY1) → Yax
  18. Line(O, ZX1) → Xax
  19. Zip(Vector[A, B], A, z_*, B, fpt_*) → fvec_*
  20. Vector(z_#, Fz1_#) → vecf_#
  21. Point(c) → A0
  22. Locus(PtAF, A0) → fLoc
  23. If(e ≟ 1, (x(z_#), y(z_#), ht), (x(fz_#), y(fz_#), ht)) → Alignfz_#
  24. Vector(z_#, Alignfz_#) → u
  25. Textfield(F) → InputBox1
  26. Locus(AF, A0) → loc1
  27. Zip((x(z), y(z)), z, fz_*) → l1
  28. Simplify(q(x, 0)) → Tf
  29. Textfield(n) → InputBox2
  30. Zip(Tf(z) + align, z, z_*) → Tfz_*
  31. Zip((x(z), y(z), ht), z, Tfz_*) → Tfpt_*
  32. Zip(Vector[A, B], A, z_*, B, Tfpt_*) → Tfvec_*
  33. Locus(TFA3, A0) → TfLoc
  34. Simplify(f(0) + (f'(0) * x) + (f''(0) / 2 * x^(2)) + (f'''(0) / 6 * x^(3)) + (f''''(0) / 24 * x^(4))) → TPf
  35. Zip(TPf(z) + align, z, z_*) → TPfz_*
  36. Zip((x(z), y(z), ht), z, TPfz_*) → TPfpt_*
  37. Zip(Vector[A, B], A, z_*, B, TPfpt_*) → TPvec_*
  38. Locus(TPFA3, A0) → TPfLoc

AI Drawing Prompt Reference

Draw the Taylor polynomials of a complex function and their mapping diagrams. Start by plotting the complex function, then calculate its Taylor polynomial terms, and plot these polynomials to show their mapping relationships with the original function. Finally, display the local magnification of the Taylor polynomial and its approximation to the original function near the origin.
The prompt above was compiled from the original drawing commands and is for reference only.