D. R. Wilton ECE Dept.

1. ECE 6382 The Steepest Descent Method D. R. Wilton ECE Dept. Adapted from original notes of Prof. David R. Jackson 8/24/10

2. Steepest Descent Method Complex Integral: We want to obtain an approximate evaluation of the integral when the real parameter Ω is very large. The functions f(z) and g(z) are analytic, so that the path Cmay be deformed if necessary. The steepest descent path (SDP) of interest will pass through a saddle point. Saddle Point:

3. X Steepest Descent Method (cont.) Path deformation: If the path does not go through a SDP, we assume that it can be deformed to do so. Any singularities encountered during the path deformation must be accounted for (e.g., residue of captured poles).

4. Steepest Descent Method (cont.) Let (Cauchy Riemann eqs.) Also, by C.R. conditions, wherever g(z) is analytic,

5. “Rubber sheet” models of u(z). Steepest Descent Method (cont.) Near the saddle point: Ignore (rotate coordinates to eliminate)

6. Steepest Descent Method (cont.) The u(x,y) function has a “saddle” shape near the SDP: Note: the saddle does not necessarily open along one of the principal axes (only when uxy(x0, y0) = 0).

7. is so large most of contribution is from z0 Steepest Descent Method (cont.) Along any descending path (where the u function decreases): Note in general, both the phase and amplitude of the integrand change. If we can find a path along which the phase does not change, the integrand will look like with most of the contribution coming from the neighborhood of z0 .

8. Steepest Descent Method (cont.) Consider a path of constant phase: This eliminates any variation in the phase term inside the integral.

9. Steepest Descent Method (cont.) u(z) landscape near z0: hill valley valley hill

10. Steepest Descent Method (cont.) u(z) and v(z) landscape near z0: Constant phase paths through z0 are also steepest descent paths (SDPs) or steepest ascent paths (SAPs)

11. Steepest Descent Method (cont.) Property: Hence C0 is either a “path of steepest descent” (SDP) or a “path of steepest ascent” (SAP).

12. y x Steepest Descent Method (cont.) proof Hence, Also, Hence

13. Steepest Descent Method (cont.) Local behavior near SP: so Let y x Compare variation of real and imag parts of g(z)- g(z0) above with “landscape” figure near the SP!

14. Steepest Descent Method (cont.) SAP: SDP:

15. Steepest Descent Method (cont.)

16. Steepest Descent Method (cont.) y u decreases SP SAP 90o u increases SDP x

17. Steepest Descent Method (cont.) SAP SDP

18. Steepest Descent Method (cont.)

19. Steepest Descent Method (cont.)

20. Steepest Descent Method (cont.) SDP Note: The direction of integration determines the sign. The “user” must determine this. SAP

21. Steepest Descent Method (cont.) Finally,

22. Example Bessel function: where Hence, we identify:

23. y x Example (cont.)

24. Example (cont.) Identify the SDP and SAP: SDP and SAP:

25. Example (cont.) SDP and SAP: SAP Examination of the u function reveals which of the two paths is the SDP. SDP

26. SDP Example (cont.) The vertical paths are added so that the original path has limits at infinity. SDP = C + Cv1 + Cv2 It is now clear which choice is correct for the departure angle:

27. Example (cont.) (ignoring the contributions of the vertical paths) Hence, so

28. Example (cont.) Hence

29. Example (cont.) Examine the path Cv1 (the path Cv2 is similar). Let

30. Example (cont.) Using repeated integration by parts:

31. Example (cont.) Hence, If we want an asymptotic expansion that is accurate to order 1/ , then the vertical paths must be considered. Otherwise, we have