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Laurent Series

Laurent Series
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2013-01-15 13:31
FunctionCalculusComplex Number
In collections Complex numbers
This example demonstrates the Laurent series expansion of the function f(z)= rac{1}{z-1}- rac{1}{z-2}. Key mathematical concepts include series expansion in complex analysis. The Sequence command is used to generate the series terms. Suitable for university mathematics students or those interested in complex analysis.

GeoGebra Commands & Parameters

  1. Circle(O, 1) → c
  2. Circle(O, 2) → d
  3. Sequence((-1 + 1 / 2^(n + 1)) * z^(n), n, 0, m) → list1
  4. Sequence(Sum[list1, n], n, 1, m - 1) → list2
  5. Sequence(Segment[Element[list2, k], Element[list2, k + 1]], k, 1, m - 1) → list3
  6. Sequence(1 / z^(n) + (z / 2)^(n) / 2, n, 0, m) → list4
  7. Sequence(Sum[list4, n] + (-1, 0), n, 1, m - 2) → list5
  8. Sequence(Segment[Element[list5, k], Element[list5, k + 1]], k, 1, m - 1) → list6
  9. Sequence((1 - 2^(n - 1)) / z^(n), n, 1, m) → list7
  10. Sequence(Sum[list7, n], n, 1, m - 2) → list8
  11. Sequence(Segment[Element[list8, k], Element[list8, k + 1]], k, 1, m - 1) → list9

AI Drawing Prompt Reference

Plot the Laurent series expansion of the function f(z) = 1/(z-1) - 1/(z-2). First, draw two circles with radii 1 and 2 centered at the origin. Then, generate a series of exponential terms, construct the series, and connect these terms with line segments to form the graphical representation of the series.
The prompt above was compiled from the original drawing commands and is for reference only.