Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 51 5The proofs are left to Problem 10. It can be shown that the orthogonal system inpart (m) is complete, so that there is a corresponding Fourier-Bessel series of orderm. This series is uniformly convergent for every function fi f (x) such that f (x)has a continuous second derivative in the interval and f (1) = 0. For proofs, seeChapter XVIII of the treatise by Watson. On page 406 of the same book it is shownthat assertion (p) can be extended to all m 2 0.The Bessel functions of the second kind are the functions Ym (x) (m = 0, 1,2, . . .)defined as follows:1 -(-l)k((im+2k XYm(x) = c - (2log + 2y - bm+k - b,k!(m + k)!k=O1 m-1--(;)-m+x (m - k - l)!k=Ok! '(7.73)where bk = 1 + 2-I + . + R-', bo = 0, and y is defined in Eq. (6.88). Thesefunctions satisfy the same differential equations (e) as the functions of first kind andhave other analogous properties. The complex-valued functionsare called Bessel functions of the third kind or Hankel functions. For details, see thebook by Watson.Tables and graphs of both Legendre polynomials and Bessel functions are available.The book by Jahnke and Emde listed at the end of the chapter provides suchdata and also a convenient summary of properties of the functions.We mention, without further discussion, several other important systems oforthogonal functions:The Jacobi polynomials.(Pjp.@'(x)]For each a! > - 1, B > - 1, the system of polynomials(n = 0, 1,2, . . .) is defined by the equationsThe functions ~$~(x) = (1 - x)fa(lj + x)i@~fp.@)(x) form a complete orthogonalsystem in the interval - 1 5 x 5 1. When a = ,!I = 0, the functions C$,(x) reduce tothe Legendre polynomials.The Hermite polynomials. The system of polynomials (H,(x)} is defined by theequations:They are orthogonal over the injinite interval -00 < x < cm with respect to the weightfunction e- f "'.