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Power Series Representations

ECE 6382. Fall 2019. David R. Jackson. Notes 7. Power Series Representations. Notes are from D . R. Wilton, Dept. of ECE. Geometric Series. Consider. Geometric Series (cont.). Generalize: ( 1  z p ):. Consider. Taylor series. Laurent series. Geometric Series (cont.). Consider.

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Power Series Representations

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  1. ECE 6382 Fall 2019 David R. Jackson Notes 7 Power Series Representations Notes are from D. R. Wilton, Dept. of ECE

  2. Geometric Series Consider

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

  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 Taylor series Laurent series Converges outside circle

  6. Uniform Convergence Consider

  7. Uniform Convergence (cont.) The series converges slower and slower as |z| approaches 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 later!

  8. Uniform Convergence (cont.) Example Consider Note: A relative error of 10-p means p significant figures of accuracy. 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.) N10 N8 N6 N4 N2 Example (cont.) Consider For example: R UsingN = 350 will give 8 significant figures everywhere inside the region. We now have uniform convergence for R = 0.95.

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

  11. The Taylor Series Expansion (cont.) Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712. Yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory. Brook Taylor (1685-1731) From Wikipedia

  12. Taylor Series Expansion of an Analytic Function (cont.)

  13. Taylor Series Expansion of an Analytic Function (cont.) Note that in the result for an, the integrand is analytic away from z0, and hence the path C is now arbitrary, as long as it encircles z0 and stays inside R. Note: It can also be shown that the series will diverge for

  14. 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.

  15. 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 converges uniformly for |z-z0| R < Rc. • 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). • 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.

  16. The Laurent Series Expansion This generalizes the concept of a Taylor series to include cases where the function is analytic in anannulus. Consider Final Result: 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 negativenallowed.) 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.

  17. The Laurent Series Expansion Consider Laurent series: The Laurent series converges inside the region The Laurent series diverges outside this region if there are singularities at

  18. The Laurent Series Expansion (cont.) The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death. Pierre Alphonse Laurent (1813 -1854) From Wikipedia

  19. The Laurent Series Expansion (cont.) This is particularly useful for functions that havepoles. Consider Examples of functions with poles, and how we can choose a Laurent series:

  20. The Laurent Series Expansion (cont.) The singularity does not have to be a simple pole:

  21. The Laurent Series Expansion (cont.) Theorem: The Laurent series expansion in the annulus region is unique. Consider (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

  22. The Laurent Series Expansion (cont.) A Taylor series is a special case of a Laurent series. Consider Here f is assumed to be analytic within C. Iff (z)is analyticwithin C, the integrand is analytic for negative values ofn. Hence, all coefficients an for negativenbecome zero (by Cauchy’s theorem).

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

  24. The Laurent Series Expansion (cont.) Contributions from the pathsc1andc2cancel! Consider Pond, island, & bridge

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

  26. The Laurent Series Expansion (cont.) Because the integrand for the coefficient is analytic with R, the path C is arbitrary as long as it stays within R. Consider We thus have The path C is now arbitrary, as long as it stays in the analytic (blue) region. where

  27. Examples of Taylor and Laurent Series Expansions Consider Use the integral formula for the an coefficients. The path C can be inside the circle or outside of it (parts (a) and (b)).

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

  29. 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 blue region.

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

  31. 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).

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

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

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

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

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

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

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

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

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

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

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

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