Special Functions. SMS 2308: Mathematical Methods

Transcription

1 Special s Special s SMS 2308: Mathematical Methods Department of Computational and Theoretical Sciences, Kulliyyah of Science, International Islamic University Malaysia. Sem /2015 Factorial Bessel s

2 Outline Factorial Bessel s for P n (x) Special s Factorial Bessel s

3 Special s x (1 x) y + [γ (α + β + 1) x] y αβy = 0, (1) where α, β and γ are fixed parameters. Regular singular points at x = 0 and x = 1. By solving this equation using Frobenius method and factorial function, a solution for this problem is obtained. Factorial Bessel s

4 Factorial Special s Factorial function (t) n is defined for n Z + {0} by (t) n := t (t + 1) (t + 2) (t + n 1), n 1. (t) 0 := 1, t 0. (2) Factorial Bessel s

5 F (α, β; γ; x) Special s F (α, β; γ; x) := 1 + or in terms of gamma function Γ F (α, β; γ; x) := Γ (γ) Γ (α) Γ (β) n=0 n=1 (α) n (β) n n!(γ) n x n. (3) Γ (α + n) Γ (β + n) x n. (4) n!γ (γ + n) Factorial Bessel s

6 Γ(x) Γ (x) := 0 Z Z Γ (x + 1) = e u u x 1+1 du = e u u x du, 0 e u u x 1 du, x > 0, (5) w = u x, 0 dv = e u du, dw = xu x 1 du, v = e u, = wv 0 Z 0 = e u u x u= u=0 = x Z 0 vdw Z e u u x 1 du = xγ (x). 0 xe u u x 1 du, Γ (x + 1) = xγ (x). (6) Special s Factorial Bessel s

7 Γ(x) Special s Γ (t + 1) = tγ (t), Γ (t + 2) = Γ ([t + 1] + 1) = (t + 1) Γ (t + 1) = (t + 1) tγ (t), Γ (t + 3) = Γ ([t + 2] + 1) = (t + 2) Γ (t + 2) = (t + 2) (t + 1) tγ (t),. Γ (t + n) = Γ ([t + n 1] + 1) = (t + n 1) (t + 2) (t + 1) tγ (t), Γ (t + n) = (t) n Γ (t). (t) n = Γ (t + n). (7) Γ (t) Factorial Bessel s

8 Bessel s of order ν Special s x 2 y + xy + ( x 2 ν 2) y = 0, (8) where ν 0 is a fixed parameter. Regular singular point at x = 0 and no other singular points in the complex plane. By solving this equation using Frobenius method, a solution for this problem is obtained. Factorial Bessel s

9 Bessel s function of the 1st kind of order ν : ( 1) n x ) 2n+ν. J ν (x) := (9) n!γ (1 + ν + n)( 2 n=0 Bessel s function of the 1st kind of order ν : J ν (x) := n=0 ( 1) n x ) 2n ν. (10) n!γ (1 ν + n)( 2 Bessel s function of the 2nd kind of order ν : Y ν (x) := cos (νπ) J ν (x) J ν (x), ν / Z. (11) sin (νπ) Neumann s function : cos (νπ) J ν (x) J ν (x) N m (x) = Y m (x) = lim. ν m sin (νπ) (12) Special s Factorial Bessel s

10 d dx [x ν J ν (x)] = x ν J ν 1 (x), (13) d ˆxν Jν (x) = d " x ν X ( 1) n «# x 2n+ν, dx dx n=0 n!γ (1 + ν + n) 2 X ( 1) n (2n + 2ν) = n=0 n!γ (1 + ν + n) 2 2n+ν x2n+2ν 1, X ( 1) n (n + ν) 2 = n=0 n! (ν + n) Γ (ν + n) 2 2n+ν x2n+2ν 1, X ( 1) n = n=0 n!γ (ν + n) 2 2n+ν 1 x(2n+ν 1)+ν, = x ν X ( 1) n «x 2n+ν 1 = x ν J ν (x). n=0 n!γ (ν + n) 2 d [ x ν J ν (x) ] = x ν J ν+1 (x), dx (14) J ν+1 (x) = 2ν x J ν (x) J ν 1 (x), (15) J ν+1 (x) = J ν 1 (x) 2J ν (x). (16) Special s Factorial Bessel s

11 for large arguments x 1: 2 J ν (x) (x πx cos νπ 2 π ), 4 2 Y ν (x) (x πx sin νπ 2 π ) (17). 4 for small arguments 0 < x 1: x ν J ν (x) 2 ν Γ (1 + ν), Y 0 (x) 2 ln x π, Y Γ (ν) 2ν ν>0 (x) πx ν. (18) Special s Factorial Bessel s

12 Special s ( 1 x 2 ) y 2xy + n (n + 1) y = 0, (19) where n is a fixed parameter. Regular singular point at x = 1. By solving this equation using power series, a solution for this problem is obtained. Factorial Bessel s

13 Special s P n (x) := 1 + k=1 ( n) k (n + 1) k k!(1) k ( ) 1 x k. (20) If we expand about x = 0, then P n takes the form [n/2] P n (x) = 2 n m=0 ( 1) m (2n 2m)! (n m)!m! (n 2m)! x n 2m, (21) where [n/2] is the greatest integer less than or equal to n/2. 2 Factorial Bessel s

14 Special s The legendre satisfy the orthogonality 1 1 P m (x) P n (x)dx = 0, n m. (22) Factorial Bessel s

15 `1 x 2 y 2xy + n (n + 1) y = 0, Special s ˆ`1 x 2 y + n (n + 1) y = 0, ˆ`1 x 2 P m + n (n + 1) Pm = 0, ˆ`1 x 2 P n + m (m + 1) Pn = 0, P n ˆ`1 x 2 P m + n (n + 1) PmP n = 0, P m ˆ`1 x 2 P n + m (m + 1) PmP n = 0, P n ˆ`1 x 2 P m + n (n + 1) PmP n P m ˆ`1 x 2 P n m (m + 1) PmP n = 0, [n (n + 1) m (m + 1)] P mp n = ˆ`1 x 2 `P np m P np m, Zx=1 x= 1 [n (n + 1) m (m + 1)] P mp ndx = [n (n + 1) m (m + 1)] Zx=1 x= 1 P mp ndx = 0. Zx=1 x= 1 Zx=1 x= 1 ˆ`1 x 2 `P np m P np m dx, P mp ndx = ˆ`1 x 2 `P np m P np m x=1 x= 1, Factorial Bessel s

16 Special s also satisfy the recurrence (n + 1) P n+1 (x) = (2n + 1) xp n (x) np n 1 (x). (23) d n P n (x) = 1 { (x 2 2 n n! dx n 1 ) } n. (24) Factorial Bessel s

17 for P n (x) Special s ( 1 2xz + z 2 ) 1/2 (25) is used as generating function for P n (x) such that ( 1 2xz + z 2 ) 1/2 = P n (x) z n, z < 1, x < 1. n=0 Factorial Bessel s