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12. Further Topics in Analysis

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 :.

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12. Further Topics in Analysis

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  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 ODE :  Hermite polynomials :

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

  7. Generating Functions Let fn(x) be a family of functions. Generating function  C encloses t = 0. gis 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 xand 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. 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 Bernoulli Polynomials Mathematica

  24. Properties of Bn (x) xboth 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 12.4. (3) Without remainder term, convergence is only asymptotic: m (3) =1.202056903... Mathematica Improvement : E-M formula starts at ns.

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