1. 12. Further Topics in Analysis
Orthogonal Polynomials
Bernoulli Numbers
Euler-Maclaurin Integration Formula
Dirichlet Series
Infinite Products
Asymptotic Series
Method of Steepest Descent
Dispersion Relations

2. 1. Orthogonal Polynomials
Rodrigues Formulas :
2nd order Sturm-Liouville ODE
with
E.g., Legendre, Hermite, Laguerre, Chebyshev, ...
Note: Bessel functions
are series.
Set where 
Coef. of xn : 

3. Self-adjoint form : with ( § 8.2 )   

4. 
 ODE : 
Rodrigues formula 
Cn = any const

5. Example 12.1.1. Rodrigues Formula for Hermite ODE
Hermite polynomials :

6. Schlaefli Integral C encloses x & f analytic on & within C. 

7. Generating Functions Let fn(x) be a family of functions.
C encloses t = 0.
g is good for deriving recurrence relations :

8. Example 12.1.2. Hermite Polynomials
Hn = Hermite polynomials 

9. Finding Generating Functions
For polynomial solutions to 2nd order Sturm-Liouville ODE
( fn = yn describable by Rodrigues formula & Schaefli integral ) :
C encloses x and w pn analytic on & within C.

10. Example 12.1.3. Legendre Polynomials
Legendre ODE :
( ODE is self-adjoint )
for Legendre
polynomials
  interchange justified if series converges

11. 
Thus, integrand is analytic for ( C lies between z & z+ ).
 z+() is outside (inside) C.  

12. Summary: Orthogonal Polynomials

13. 2. Bernoulli Numbers Bn = Bernoulli numbers
Caution: Definition not unique.
n  1   

14.   

15. Recursion Relation for Bn 

16. 
m = 2,3, ... 
Let
m even  
m odd  

17. Values of B2n
Mathematica

18. Another Generating Function 

19. Contour Integral Representation
 analytic near z = 0. 
C encloses 0 but no other poles
E.g. :
Bn : rather tedious

20. Better Contour   

21. Caution : another often used definition is
Mathematica
Caution : another often used definition is
Number theory :
von Staudt-Clausen theorem
E.g.

22. Miscellaneous Usages of Bn
In sums :
In series expansions :
e.g., tanx, cotx, ln|sinx|, sin1x, ln|tanx|, cosh 1x, tanhx, cothx, etc

23. Bernoulli Polynomials
Mathematica

24. Properties of Bn (x)
x both sides : 
x = 1 : 

25. 3. Euler-Maclaurin Integration Formula
Consider
 n  1

26.  n  1
n = 0 is a special case since B1  1/2  0. 
Euler-Maclaurin integration formula

27. Euler-Maclaurin integration formula 
Approximate sum by integral

28. Example 12.3.1. Estimation of (3)

29. Table (3)
Without remainder term, convergence is only asymptotic: m
(3) =
Mathematica
Improvement : E-M formula starts at ns .