1 Asymptotic Series ECE 6382 David R. Jackson

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

Asymptotic Series (cont.) 3 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.

Big O and small o notation 4 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. Official definition: There exists a constant k and a radius R such that For all

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

6 This notation is helpful for defining and discussing asymptotic series. Small o notation: Qualitatively, this means that f “is smaller than” g as z gets large. Official definition: For any  there exists a radius R (which depends on  ) such that For all Big O and small o notation (cont.)

7 Examples:

8 Definition of Asymptotic Series Definition 1: Definition 2: Equivalence of definitions: Assume Def. 1: Assume Def. 2:

9 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 = z 0. General “rule of thumb”: Pick N so that the N th term in the series is the smallest.

10 Generation of Asymptotic Series  Asymptotic methods such as the method of steepest descent.  Integration by parts (useful when there is no saddle point).  Other specialized techniques. Various method can be used to generate an asymptotic series expansion of a function.

11 Example The exponential integral: z z X Re (t) Im (t) z Branch cut Re (z) Im (z) Note: E 1 (z) is discontinuous across the negative real axis.

12 Example (cont.) Use integration by parts:

13 Example (cont.) Using continued integration by parts ( N times): or Question: Is this a valid asymptotic series? Note: a 0 = 0 here.

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

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

16 Example (cont.) n = odd n = even Exact value = nAnAn Using n = 5 or 6 is optimum for x = 5. This is where | A n | is the smallest.

17 Example (cont.) As x gets large, the error in stopping with N terms is approximately given by the first term that is omitted. Hence (from Def. 1) Therefore, we have Hence

18 Example (cont.) The first neglected term in the series (called T(x )) is shown here. N = 1 N = 2 x |T(x)|

19 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: The series represents an analytic and (hence continuous) function that approaches a constant as z  . Recall: If a power series converges, it represents an analytic function.

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