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

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 : 

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

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

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

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

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

Example 12.1.2. Hermite Polynomials Hn = Hermite polynomials 

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.

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

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

Summary: Orthogonal Polynomials

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

  

Recursion Relation for Bn  

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

Values of B2n Mathematica

Another Generating Function  

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

Better Contour   

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

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

Bernoulli Polynomials Mathematica

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

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

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

Euler-Maclaurin integration formula   Approximate sum by integral

Example 12.3.1. Estimation of (3) 

Table 12.4. (3) Without remainder term, convergence is only asymptotic: m (3) =1.202056903... Mathematica Improvement : E-M formula starts at ns .