1. 4.Dirichlet Series Dirichlet series : E.g.,(Riemann) Zeta function

2. Example 12.4.1.Evaluation of  (2) Example 11.9.1: 

3. Using B 2n § 12.2 :  Mathematica s

4. Theorem : If 0  |a n | < 1, then converges / diverges if converges / diverges. 5.Infinite Products Infinite product :  Proof :   Comparison test  S & ln | P | have same convergency. QED.

5. Example 12.5.1.Convergence of Infinite Products for sin z & cos z § 11.7 :  product converges  finite z.

6. Example 12.5.2.An Interesting Product  

7. 6.Asymptotic Series We’ll consider only two types of integrals that lead to asymptotic series

8. Exponential Integral Exponential integral  Set see § 13.6 Alternatively, let

9. Ratio test : series diverges  x. Set

10.  x  0  R n alternates in sign  so does s n

11. R n alternates in sign  so does s n Mathematica See § 13.6

12. Cosine & Sine Integrals Sine integral Cosine integral  Mathematica  CC

13.   

14. Definition of Asymptotic Series then is the asymptotic expansion of f (x). Partial sum Let If but Poincare :   but Asymptotic expansion is unique if exists.

15. 7.Method of Steepest Descent Method of steepest descent / Saddle point method is used to evaluate asymptotically integrals of the form where C can be deformed to pass through a saddle point z 0 of the analytic function f so that  |z| Re f is most negative along C. for t >> 1   = angle of C at z 0.

16. Let z 0 be a saddle point of f (z) Saddle Point Method  ( true for any C through z 0 ) most negative is the steepest decent away from z 0. On C near z 0 :

17.   Preferred path :  r Re f is most negative along r (steepest descent ).  whereupon |  r Im f | = 0, i.e., Im f = const. Choice compensates sense of C. 

18. Example 12.7.1.Asymptotic Form of the Gamma Function      Sterling’s formula 

19. Example 12.7.2.Saddle Point Method Avoids Oscillations     

20. 8.Dispersion Relations Let f (z) be analytic in upper half z -plane ( for Im z  0 ) where z 0 in upper plane : x 0 on x -axis :   Kramer-Kronig (dispersion) relations

21. && Symmetry Relations Let  crossing conditions    Sum Rules Ex.12.8.4

22. For poor conductor  Optical Dispersion ~ EM wave moving toward +x with In a media with dielectric constant ,  = 1 & conductivity  (Gaussian units) :  ~ absorption  In general,

23.  1 as    Let  Kramer-Kronig relations

24. Parseval relation : is finite for any transform pair u & v. The Parseval Relation Hilbert transform :  u & v are Hilbert transforms of each other. Proof :

25. Ex.12.8.8   Thus, presence of refraction ( Re n 2  1 ) for some  range  absorption ( Im n 2  0 ) for some  range.