Mathematical Methods for Physics and Engineering - Matematica.NET

18.13 EXERCISES[ You will find it convenient to use∫ ∞x 2n e −x2 dx = (2n)!√ π−∞2 2n n!for integer n ≥ 0. ]18.9 By initially writing y(x) asx 1/2 f(x) and then making subsequent changes ofvariable, reduce Stokes’ equation,d 2 y+ λxy =0,dx2 to Bessel’s equation. √Hence show that a solution that is finite at x =0isamultiple of x 1/2 J 1/3 ( 2 λx3 ).318.10 By choosing a suitable form for h in their generating function,[ ( zG(z,h) =exp h − 1 )] ∞∑= J n (z)h n ,2 hn=−∞show that integral repesentations of the Bessel functions of the first kind aregiven, for integral m, by∫ 2πJ 2m (z) = (−1)m cos(z cos θ) cos2mθ dθ m ≥ 1,π 0∫ 2πJ 2m+1 (z) = (−1)m+1 cos(z cos θ) sin(2m +1)θdθ m≥ 0.π 018.11 Identify the series for the following hypergeometric functions, writing them interms of better known functions:(a) F(a, b, b; z),(b) F(1, 1, 2; −x),(c) F( 1 , 1, 3 ; 2 2 −x2 ),(d) F( 1 , 1 , 3 ; 2 2 2 x2 ),(e) F(−a, a, 1 2 ;sin2 x); this is a much more difficult exercise.18.12 By making the substitution z =(1− x)/2 and suitable choices for a, b and c,convert the hypergeometric equation,z(1 − z) d2 udu+[c − (a + b +1)z ] − abu =0,dz2 dzinto the Legendre equation,(1 − x 2 ) d2 y dy− 2x + l(l +1)y =0.dx2 dxHence, using the hypergeometric series, generate the Legendre polynomials P l (x)for the integer values l =0, 1, 2, 3. Comment on their normalisations.18.13 Find a change of variable that will allow the integral∫ ∞√u − 1I =1 (u +1) du 2to be expressed in terms of the beta function, and so evaluate it.18.14 Prove that, if m and n are both greater than −1, then∫ ∞u mI =0 (au 2 + b) du = Γ[ 1 (m 2 +1)]Γ[1 (n +1)]2 (m+n+2)/2 2a (m+1)/2 b (n+1)/2 Γ[ 1 (m + n +2)].2643