1. Power Series Representations ECE 6382 Notes are from D. R. Wilton, Dept. of ECE David R. Jackson 1

2. Geometric Series Consider 2 1 1 1

3. Geometric Series (cont.) Consider Generalize: ( 1  z 0 ): 3

4. Consider Decide whether |z 0 -z ’ | or |z-z ’ | is larger (i.e., if z is inside or outside the circle at right), and factor out the term with largest magnitude! Geometric Series (cont.) 4

5. Consider Geometric Series (cont.) 5 Converges inside circle Converges outside circle Summary

6. Uniform Convergence Consider 6

7. Key Point: Term-by-term integration of a series is allowed over any region where it is uniformly convergent. We use this property extensively in the following! 7 Uniform Convergence (cont.) Non-uniform convergence Uniform convergence 1 1 1 1 1 R

8. Uniform Convergence (cont.) Consider 8 The closer z gets to the boundary of the circle, the more terms we need to get the same level of accuracy (non-uniform convergence).

9. Consider N1N1 N8N8 N6N6 N4N4 N2N2 9 Uniform Convergence (cont.) For example: 1 1 R Using N = 350 will give 8 significant figures everywhere inside the region.

10. Taylor Series Expansion of an Analytic Function 10

11. 11 Taylor Series Expansion of an Analytic Function (cont.) Note: It can be shown that the series will diverge for

12. 12 Taylor Series Expansion of an Analytic Function (cont.) The radius of convergence of a Taylor series is the distance out to the closest singularity. Summary of Result Key point: The point z 0 about which the expansion is made is arbitrary, but It determines the region of convergence of the Laurent or Taylor series.

13. 13 Taylor Series Expansion of an Analytic Function (cont.) Properties of Taylor Series  A Taylor series will converge for |z-z 0 | < R.  A Taylor series will diverge for |z-z 0 | > R.  A Taylor series may be differentiated or integrated term-by-term within the radius of convergence. This does not change the radius of convergence.  A Taylor series converges absolutely inside the radius of convergence (i.e., the series of absolute values converges).  A Taylor series converges uniformly inside the radius of convergence.  When a Taylor series converges (inside the radius of convergence) the resulting function is an analytic function.  Within the common region of convergence, we can add and multiply Taylor series, collecting terms to find the resulting Taylor series. R = radius of convergence = distance to closest singularity J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9 th Ed., McGraw-Hill, 2013.

14. The Laurent Series Expansion Consider This generalizes the concept of a Taylor series to include cases where the function is analytic in an annulus. or Converges for 14 z0z0 a b z zaza zbzb Here z a and z b are two singularities.

15. The Laurent Series Expansion (cont.) Consider Examples: This is particularly useful for functions that have poles. But the expansion point z 0 does not have to be at a singularity, nor must the singularity be a simple pole: z0z0 a b z zbzb zaza 15 y x Branch cut Pole

16. Consider z 0 = 0 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 any series of valid steps, it is correct.) Hence Example: 16 The Laurent Series Expansion (cont.) This is justified by our Laurent series expansion formula, derived later.

17. 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.) 17 The Laurent Series Expansion (cont.)

18. Pond, island, & bridge Pond: Domain of analyticity Island: Region containing singularities Bridge: Region connecting island and boundary on pond 18 The Laurent Series Expansion (cont.) We use the “bridge” principle again

19. Consider Contributions from the paths c 1 and c 2 cancel! Pond, island, & bridge 19 The Laurent Series Expansion (cont.)

20. Consider 20 The Laurent Series Expansion (cont.) We thus have

21. Examples of Taylor and Laurent Series Expansions Consider 21 Hence

22. Consider 22 Examples of Taylor and Laurent Series Expansions (cont.) Hence

23. Consider 23 Examples of Taylor and Laurent Series Expansions (cont.) Often it is easier to directly use the geometric series formula together with some algebra, instead of the contour integral approach, to determine the coefficients of the Laurent expanson. This is illustrated next (using the same example as in Example 1).

24. Consider 24 Examples of Taylor and Laurent Series Expansions (cont.) Hence

25. Consider 25 Examples of Taylor and Laurent Series Expansions (cont.) Hence Alternative expansion:

26. Consider 26 Examples of Taylor and Laurent Series Expansions (cont.) Hence (Taylor series)

27. Consider 27 Examples of Taylor and Laurent Series Expansions (cont.) (Laurent series) so

28. Consider 28 Examples of Taylor and Laurent Series Expansions (cont.) Hence

29. Consider 29 Examples of Taylor and Laurent Series Expansions (cont.)

30. Consider 30 Examples of Taylor and Laurent Series Expansions (cont.)

31. Summary of Methods for Generating Taylor and Laurent Series Expansions Consider 31

32. Summary of Methods for Generating Taylor and Laurent Series Expansions (cont.) Consider 32