Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONSsingular points, whereas any singular point not satisfying both these criteria istermed an irregular or essential singularity.◮Legendre’s equation has the form(1 − z 2 )y ′′ − 2zy ′ + l(l +1)y =0, (16.8)where l is a constant. Show that z =0is an ordinary point and z = ±1 are regular singularpoints of this equation.Firstly, divide through by 1 − z 2 to put the equation into our standard form (16.7):y ′′ −2z l(l +1)1 − z 2 y′ +1 − z y =0.2Comparing this with (16.7), we identify p(z) andq(z) asp(z) =−2z1 − z = −2zl(l +1) l(l +1), q(z) = =2 (1 + z)(1 − z) 1 − z 2 (1 + z)(1 − z) .By inspection, p(z) andq(z) are analytic at z = 0, which is therefore an ordinary point,but both diverge for z = ±1, which are thus singular points. However, at z =1weseethat both (z − 1)p(z) and(z − 1) 2 q(z) are analytic and hence z = 1 is a regular singularpoint. Similarly, at z = −1 both (z +1)p(z) and(z +1) 2 q(z) are analytic, and it too is aregular singular point. ◭So far we have assumed that z 0 is finite. However, we may sometimes wish todetermine the nature of the point |z| →∞. This may be achieved straightforwardlyby substituting w =1/z into the equation and investigating the behaviour atw =0.◮Show that Legendre’s equation has a regular singularity at |z| →∞.Letting w =1/z, the derivatives with respect to z becomedydz = dy dwdw dz = − 1 dy dy=z 2 −w2dw dw ,d 2 ydz = dw ( ) (d dy= −w 2 −2w dy ) (2 dz dw dzdw − d2 yw2 = w 3 2 dy )dw 2 dw + w d2 y.dw 2If we substitute these derivatives into Legendre’s equation (16.8) we obtain(1 − 1 ) (w 3 2 dy )w 2 dw + w d2 y+2 1 dydw 2 w2 + l(l +1)y =0,w dwwhich simplifies to givew 2 (w 2 − 1) d2 y dy+2w3 + l(l +1)y =0.dw2 dwDividing through by w 2 (w 2 − 1) to put the equation into standard form, and comparingwith (16.7), we identify p(w) andq(w) asp(w) =2wl(l +1), q(w) =w 2 − 1 w 2 (w 2 − 1) .At w =0,p(w) is analytic but q(w) diverges, and so the point |z| →∞is a singular pointof Legendre’s equation. However, since wp and w 2 q are both analytic at w =0,|z| →∞is a regular singular point. ◭534