Asymptotic Series

1. ECE 6382 Fall 2019 David R. Jackson Notes 13 Asymptotic Series

2. Asymptotic Series An asymptotic series (as z ) is of the form or Note the “asymptotically equal to” sign. The asymptotic series shows how the function behaves as z gets large in magnitude. Important point: An asymptotic series does not have to be a converging series. (This is why we do not use an equal sign.)

3. Asymptotic Series (cont.) Properties of an asymptotic series: • For a fixed number of terms in the series, the series get more accurate as the magnitude of z increases. • For a fixed value of z, the series does not necessarily get more accurate as the number of terms increases. • The series does not necessarily converge as we increase the number of terms, for a fixed value of z. Note: We can also talk about

4. Big O and small o notation This notation is helpful for defining and discussing asymptotic series. Big O notation: Qualitatively, this means that f “behaves like” g as z gets large (or possibly goes to zero even faster). Definition: There exists a constant k and a radius R such that For all

5. Big O and small o notation (cont.) Examples:

6. Big O and small o notation (cont.) Small o notation: Qualitatively, this means that f “gets smaller than” g as z gets large. Definition: For any there exists a radius R (which depends on )such that For all

7. Big O and small o notation (cont.) Examples:

8. Definition of Asymptotic Series Definition of asymptotic series: In order for this to be an asymptotic series we require the following: For anyN As z gets large, the error in stopping at term n = Nis smaller than this last term. Example:

9. Definition of Asymptotic Series Theorem If Then

10. Definition of Asymptotic Series Proof of theorem Assume (from definition of asymptotic series)

11. Summing Asymptotic Series • One must be careful when summing an asymptotic series, since it may diverge: it is not clear what the optimum number of terms is, for a given value of z= z0. General “rule of thumb”: Pick N so that the Nth term in the series is the smallest. (See the example later.)

12. Generation of Asymptotic Series Various method can be used to generate an asymptotic series expansion of a function. • Integration by parts • The method of steepest descent • Watson’s Lemma • Other specialized techniques

13. Example The exponential integral function: Branch cut X Note:E1(z) is discontinuous (by 2i) across the negative real axis.

14. Example (cont.) Use integration by parts: Note: It is very important which of the two functions is chosen to be u and which one is chosen to be v.

15. Example (cont.) Using integration by parts N times: “Error term” or Question: Is this a valid asymptotic series: Note:a0 = 0 here.

16. Example (cont.) Examine the difference term: or so

17. Example (cont.) Hence Question: Is this a converging series? Use the d’Alembert ratio test: The series diverges!

18. Example (cont.) n= odd Exact value F(5) = 0.1704 n= even Using n = 5 or 6 is optimum for x = 5. This is also where |An| is the smallest.

19. Example (cont.) As x gets large, the error in stopping with N terms is approximately given by thefirst term that is omitted. To see this, use: (from definition of asymptotic series) Therefore, we have (separating out the last term from the sum) Hence

20. Example (cont.) N = 1 N = 2 Note: Using more terms is not better for small x!

21. Note on Converging Series Assume that a series converges for all z 0, so that Then it must also be a valid asymptotic series: Proof:

22. Note on Converging Series This is a converging series that approaches a constant (the value aN+1) as z gets large. (It is also a converging Taylor series if we let w = 1/z, and hence an analytic function of w.) Hence, we have:

23. Note on Converging Series (cont.) Example: The point z = 0 is an isolated essential singularity, and there are no other singularities out to infinity. This Laurent series converges for all z  0. This is a valid asymptotic series.