1. ECE 6382 Notes 6 Power Series Representations Fall 2016
David R. Jackson
Notes 6
Power Series Representations
Notes are from D. R. Wilton, Dept. of ECE

2. Geometric Series 1
Consider

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

4. Geometric Series (cont.)
Consider
Decide whether |zp-z0| or |z-z0| is larger (i.e., if z is inside or outside the circle at right), and factor out the term with largest magnitude!

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

6. Uniform Convergence
Consider

7. Uniform Convergence (cont.)
The series converges slower and slower as |z| approaches 1. 1 1 R
Non-uniform convergence
Uniform convergence
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!

8. Uniform Convergence (cont.)
Consider
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. Uniform Convergence (cont.)
Consider
For example: N1 N8 N6 N4 N2 1 R
Using N = 350 will give 8 significant figures everywhere inside the region.

10. The Taylor Series Expansion
This expansion assumes we have an analytic function.
Consider z0 Rc z zs
“derivative formula”
Here zs is the closest singularity.
Note: both forms are useful.
The path C is any counterclockwise closed path that encircles the point z0.
Rc = radius of convergence of the Taylor series
The Taylor series will converge within the radius of convergence, and diverge outside.

11. Taylor Series Expansion of an Analytic Function

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

13. Taylor Series Expansion of an Analytic Function (cont.)
The radius of convergence of a Taylor series is the distance out to the closest singularity.
Key point:
The point z0 about which
the expansion is made is arbitrary, but
It determines the region of convergence
of the Taylor series.

14. Taylor Series Expansion of an Analytic Function (cont.)
Properties of Taylor Series
Rc = radius of convergence = distance to closest singularity
A Taylor series will converge for |z-z0| < Rc (i.e., inside the radius of convergence).
A Taylor series will diverge for |z-z0| > Rc (i.e., outside the radius of convergence).
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 for |z-z0| < R < Rc.
When a Taylor series converges 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.
J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9th Ed., McGraw-Hill, 2013.

15. The Laurent Series Expansion
This generalizes the concept of a Taylor series to include cases where the function is analytic in an annulus.
Consider z0 a b z za zb or
Here za and zb are two singularities.
(derived later)
Note:
The point zb may be at infinity.
(This is the same formula as for the Taylor series, but with negative n allowed.)
The path C is any counterclockwise closed path that stays inside the annulus an encircles the point z0.
Note: We no longer have the “derivative formula” as we do for a Taylor series.

16. The Laurent Series Expansion (cont.)
This is particularly useful for functions that have poles.
Consider z0 a b z zb za
Examples of functions with poles: y x
Branch cut
Pole
But the expansion point z0 does not have to be at a
singularity, nor must the singularity be a simple pole:

17. The Laurent Series Expansion (cont.)
Theorem:
The Laurent series expansion in the annulus region is unique.
Consider
z0 = 0 a b X x y
(So it doesn’t matter how we get it; once we obtain it by any series of valid steps, it is correct!)
This is justified by our Laurent series expansion formula, derived later.
Example:
Hence

18. The Laurent Series Expansion (cont.)
Consider
A Taylor series is a special case of a Laurent series. z0 a b C
Here f is analytic within C.
If f (z) is analytic within C, the integrand is analytic for negative values of n. Hence, all coefficients an for negative n become zero (by Cauchy’s theorem).

19. The Laurent Series Expansion (cont.)
Derivation of Laurent Series
We use the “bridge” principle again
David, this is a Google maps shot of the “engineering pond” outside our building!
Pond, island, & bridge
Pond: Domain of analyticity
Island: Region containing singularities
Bridge: Region connecting island and boundary of pond

20. The Laurent Series Expansion (cont.)
Contributions from the paths c1 and c2 cancel!
Consider
Pond, island, & bridge

21. The Laurent Series Expansion (cont.)
Consider
We thus have
Let C1  C2.

22. Examples of Taylor and Laurent Series Expansions
Consider
Use the integral formula for the an coefficients.
The path C can be inside the yellow region or outside of it.

23. Examples of Taylor and Laurent Series Expansions
Consider
From uniform convergence
From previous example in Notes 3
Hence
The path C is inside the yellow region.

24. Examples of Taylor and Laurent Series Expansions (cont.)
Consider
From previous example in Notes 3
From uniform convergence
Hence
The path C is outside the yellow region.

25. Examples of Taylor and Laurent Series Expansions (cont.)
Consider
Summary of results for the example:

26. Examples of Taylor and Laurent Series Expansions (cont.)
Consider
Note:
Often it is easier to directly use the geometric series (GS) 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).

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

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

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

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

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

32. Examples of Taylor and Laurent Series Expansions (cont.)
Summary of results for example
Consider

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

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

35. Examples of Taylor and Laurent Series Expansions (cont.)
Consider
The branch cut is chosen away from the yellow region.

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

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