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

Geometric Series 1 Consider

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

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!

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

Uniform Convergence Consider

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!

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).

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.

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.

Taylor Series Expansion of an Analytic Function

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

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.

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.

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.

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:

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

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).

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

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

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

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.

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.

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.

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

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).

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

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

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

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

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

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

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

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

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

Summary of Methods for Generating Taylor and Laurent Series Expansions Consider

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