1. D. R. Wilton ECE Dept. ECE 6382 Power Series Representations 8/24/10

2. Geometric Series Consider 1 1

3. Geometric Series, cont’d Consider

4. Geometric Series, cont’d Consider Factor out the largest term!

5. Uniform Convergence Consider

6. Uniform Convergence, cont’d Consider

7. Uniform Convergence, cont’d Consider N1N1 N8N8 N6N6 N4N4 N2N2 1 1 Key Point: Term-by-term integration of a series is allowed over any region where it is uniformly convergent.

8. Taylor Series Expansion of an Analytic Function

9. Taylor Series Expansion of an Analytic Function, Cont’d

10. The Laurent Series Expansion Consider This generalizes the concept of a Taylor series, to include cases where the function is analytic in an annulus. z0z0 a b or Converges for z Key point: The point z 0 about which the expansion is made is arbitrary, but determines the region of convergence of the Laurent or Taylor series. zaza zbzb

11. The Laurent Series Expansion, cont’d Consider Examples: z0z0 a b This is particularly useful for functions that have poles. z But the expansion point z 0 does not have to be at a singularity, nor must the singularity be a simple pole: y x branch cut pole zbzb zaza

12. The Laurent Series Expansion, cont’d Consider z0z0 a b Theorem: The Laurent series expansion in the annulus region is unique. (So it doesn’t matter how we get it; once we obtain it by valid steps, it must be correct.) Hence Example:

13. The Laurent Series Expansion, cont’d Consider We next develop a general method for constructing the coefficients of the Laurent series. z0z0 a b C Note: If f ( z ) is analytic at z 0, the integrand is analytic for negative values of n. Hence, all coefficients for negative n become zero (by Cauchy’s theorem). Final result: (This is the same formula as for the Taylor series, but with negative n allowed.)

14. Consider The Laurent Series Expansion, cont’d Pond, island, & bridge

15. Consider Contributions from the paths c 1 and c 2 cancel! The Laurent Series Expansion, cont’d Pond, island, & bridge

16. The Laurent Series Expansion, cont’d Consider

17. Examples of Taylor and Laurent Series Expansions Consider

18. Examples of Taylor and Laurent Series Expansions,cont’d Consider

19. Examples of Taylor and Laurent Series Expansions,cont’d Consider

20. Examples of Taylor and Laurent Series Expansions,cont’d Consider

21. Examples of Taylor and Laurent Series Expansions,cont’d Consider

22. Examples of Taylor and Laurent Series Expansions,cont’d Consider

23. Examples of Taylor and Laurent Series Expansions,cont’d Consider

24. Examples of Taylor and Laurent Series Expansions,cont’d Consider

25. Summary of Methods for Generating Taylor and Laurent Series Expansions Consider