1. Rodrigues Formula :Laguerre Polynomials ( n! changes scale ) 3. Laguerre Functions Laguerre ODE C encircles x but no other singularities Schlaefli integral :  Hermitian form

2. Generating Function  

3. Properties of L n (x)  

4.  

5.   

6. Eliminate g     as before

7. Table & Figure.Laguerre Polynomials Mathematica  L n orthogonal over [0,  ]

8. Power Series  

9. Orthonormality  orthogonality Ex.18.3.3 Set  

10. Associated Laguerre Polynomials   Alternative definition:

11. Generating Function Proof :   (1+t) both sides :  Gives L n k with k  0 only. i.e., only terms n  l are used.

12.     g l is a correct generating function for L n k.  g l has correct scale.

13.   same as before  

14. More Recurrence   

15. ODE  Associated Laguerre eq.

16.  Hermitian form Orthogonality obtained from

17. Set  Rodrigues formula (re-scaled by n! ) Laguerre functions  Mathematica Set   For non-integer n, solutions to ODE are not polynomials & diverge as x k e x.

18. Example 18.3.1. The Hydrogen Atom Schrodinger eq. for H-like atom of atomic number Z. SI units  B.C. for bound states : R(0) finite & R(  ) = 0. Let 

19.   with  

20. Integers   1 must be an integer. Set  Bohr radius   

21. 4.Chebyshev Polynomials Ultraspherical polynomials (Gegenbauer polynomials)  = ½  Legendre polynomials  = 0 (1)  Type I (II) Chebyshev (Tchebycheff / Tschebyschow ) polynomials Type II Polynomials U n : Application: 4-D spherical harmonics in angular momentum theory.

22. Type I Polynomials T n (x)  = 0 : LHS = 1. Remedy:  where  = 0 :  Set

23. Recurrence  Similarly

24.  Other recurrence :

25. Table & Figure Mathematica

26. ODEs unuable Better choice is Proof :  Rodrigues formula c n (  ) = scaling constant

27. Special Values Ex.18.4.1-2   Rodrigues formula   0  0   1  1

28. Trigonometric Form     

29.    

30. 

31. From g (  ) or ODE (Frobenius series) :

32.  at at Application to Numerical Analysis Let If error decreases rapidly for m > M, then  error satisfies the minimax principle. i.e., max of error is minimized by spreading it into regions between points of negligible error.

33. Example 18.4.1.Minimizing the Maximum Error Mathematica max |  f | is smallest for T k expansion 4-term expansions ( k max = 3 )

34. Orthogonality   0 :   1 : Normalization obtained using trigonometric form