1. Prof. David R. Jackson ECE Dept. Spring 2014 Notes 32 ECE 6341 1

2. Laplace’s Method Consider Assumptions: Note: If there is no point where the derivative of the g function vanishes, then integration by parts may be used, in exactly the same manner as was done for the case when there was a j in the exponent (Notes 28). 2 (“saddle point”)

3. Laplace’s Method (cont.) 3

4. The exponential function behaves similar to a  function as   . 4

5. Laplace’s Method (cont.) 5 Hence

6. Laplace’s Method (cont.) Hence Since only the local neighborhood of x 0 is important, we can write Next, let 6

7. Laplace’s Method (cont.) Next use Then Hence 7

8. Laplace’s Method (cont.) Summary 8

9. Complete Asymptotic Expansion (Using Watson’s Lemma) Let We adopt the convention of positive and negative s as shown in the figure, in order to make the mapping x ( s ) unique. 9

10. Complete Asymptotic Expansion (cont.) Let 10

11. Assume (This is Watson’s Lemma.) Then Complete Asymptotic Expansion (cont.) 11

12. Since only the local neighborhood of s = 0 is important, Because of symmetry, Complete Asymptotic Expansion (cont.) (The error made in extending the limits is exponentially small.) 12

13. Denote Use Complete Asymptotic Expansion (cont.) 13 Then we have

14. Now use Hence Complete Asymptotic Expansion (cont.) 14

15. Hence Recall that Complete Asymptotic Expansion (cont.) 15

16. Summary Complete Asymptotic Expansion (cont.) Note: The hard part is determining the a n coefficients! 16 Assume Then Note: Integer powers are assumed, since h(s) is assumed to be analytic

17. Leading term: so that Complete Asymptotic Expansion (cont.) 17 Note

18. Recall that and that To find h (0), we must evaluate the derivative term. To do this, we take the derivative with respect to x : Note: At s = 0 ( x = x 0 ), this yields 0 = 0 (not useful). Complete Asymptotic Expansion (cont.) 18

19. At Take one more derivative: Complete Asymptotic Expansion (cont.) 19

20. Hence We then have Therefore Complete Asymptotic Expansion (cont.) 20

21. or Complete Asymptotic Expansion (cont.) 21 This agrees with the result from Laplace’s method.

22. Watson’s Lemma (Alternative Form) Here we do not necessarily assume integer powers in the expansion of the function, and we also start the integral at s = 0. Assume This one-sided form occurs when integrating along branch cuts in the complex plane (discussed later). 22

23. Watson’s Lemma (Alternative Form) (cont.) Then Let (This is Watson’s Lemma.) Assume 23

24. Hence Watson’s Lemma (Alternative Form) (cont.) 24

25. Summary Watson’s Lemma (Alternative Form) (cont.) 25 Assume Then