bess_orthog

Physics 116C The orthogonality relation satisfied by Bessel functions Peter Young (Dated: October 22, 2009) We showed in class that the Bessel function Jp(x) satisfies the following differential equation (Bessel’s equation) x2d2Jp dx2+xdJp dx+ (x2−p2)Jp= 0. (1) which can be written as xd dx/parenleftbigg xdJp dx/parenrightbigg + (x2−p2)Jp= 0. (2) The variable pneed not be an integer. It turns out to be useful to define a new v ariable tbyx=at, where ais a constant which we will take to be a zero of Jp, i.e.Jp(a) = 0. Let us define u(t) =Jp(at), (3) which implies u(1) = 0 , (4) and substituting into Eq. (2) gives td dt/parenleftbigg tdu dt/parenrightbigg + (a2t2−p2)u= 0, (5) since xd/dx is equivalent to td/dt. We can also write down the equation obtained by picking another zero, bsay. Defining v(t) =Jp(bt) so v(1) = 0 , (6) we have td dt/parenleftbigg tdv dt/parenrightbigg + (b2t2−p2)v= 0. (7) To derive the orthogonality relation, we multiply Eq. (5) by v, and Eq. (7) by u. Subtracting and dividing by tgives vd dt/parenleftbigg tdu dt/parenrightbigg −ud dt/parenleftbigg tdv dt/parenrightbigg + (a2−b2)tuv= 0. (8) 2 The first two terms in Eq. (8) can be combined as d dt/parenleftbigg v tdu dt−u tdv dt/parenrightbigg , (9) since the extra terms present in Eq. (9), but not in Eq. (8), wh en the derivatives are expanded out are equal and opposite and so cancel. Hence we have d dt/parenleftbigg v tdu dt−u tdv dt/parenrightbigg + (a2−b2)t u v= 0. (10) We next integrate this over the range of tfrom 0 to 1, which gives /bracketleftbigg v tdu dt−u tdv dt/bracketrightbigg1 0+ (a2−b2)/integraldisplay1 0t u(t)v(t)dt= 0. (11) The integrated term vanishes at the lower limit because t= 0, and it also vanishes at the upper limit because u(1) = v(1) = 0, see Eqs. (4) and (6). Hence, if a/negationslash=b, Eq. (11) gives /integraldisplay1 0t u(t)v(t)dt= 0, (12) which, using Eqs. (3) and (6), can be written /integraldisplay1 0t Jp(at)Jp(bt)dt= 0. (13) This is the desired orthogonality equation. Remember we req uire that aandbare distinct zeroes ofJp, so both Bessel functions in Eq. (13) vanish at the upper limi t. Ifa=byou showed in a homework problem that the corresponding inte gral is given by /integraldisplay1 0t J2 p(at)dt=1 2J′2 p(a), (14) where J′ p(a)≡dJp(x)/dx|x=a. This can be written in different ways. From Eq. (10) of the han dout on “The differential equation satisfied by Bessel functions” we have J′ p(a) =Jp−1(a) (remember thatJp(a) = 0), and from Eq. (11) we have J′ p(a) =−Jp+1(a). Hence /integraldisplay1 0t J2 p(at)dt=1 2J′2 p(a) =1 2J2 p−1(a) =1 2J2 p+1(a), (15) where ais a zero of Jp(x), i.e. Jp(a) = 0. We are now able to expand a given function f(x) in the interval from zero to 1 (provided f(1) = 0) as a Bessel series f(x) =/summationdisplay mamJp(cmpx), (16) 3 where cmpis the m−th zero of the Bessel function Jp(x). Note that p, the order of the Bessel function, is fixed in Eq. (16). Equation (16) will be very usef ul when solving partial differential equations with certain boundary conditions. Multiplying E q. (16) by xJp(cnpx), integrating from 0 to 1, and using Eqs. (13) and (15), the coefficient anis easily seen to be an=1 1 2J′2p(cnp)/integraldisplay1 0x f(x)Jp(cnpx)dx . (17) From Eq. (15) we see that J′ p(cnp) in the denominator can be replaced by Jp−1(cnp) orJp+1(cnp). Bessel series are analogous to Fourier series and Legendre s eries that we have met before. As we shall discuss in class and in more detail in the homework, Bes sel functions, Legendre polynomials, and sines and cosines, are just particular examples of sets o f functions which solve a general class of differential equations known as “Sturm-Liouville”. Sturm Liouville equations are important because: All equations of the Sturm-Liouville type have an orthogona lity property which permits a given function defined over an appropriate range and with ap propriate boundary conditions to be expressed as a linear combination of their s olutions, in which each coefficient can be determined simply by doing an integral. You will investigate Sturm-Liouville equations in a problem in the next homework assignment.