Orthogonality of Bessel Functions - USM

1 Jim LambersMAT 415/515 Fall Semester 2013-14 Lecture 15 NotesThese notes correspond to Section in the of Bessel FunctionsSince Bessel Functions often appear in solutions of PDE, it is necessary to be able to computecoefficients of series whose terms include Bessel Functions . Therefore, we need to understand theirorthogonality the Bessel equation 2d2J (k )d 2+ dJ (k )d + (k2 2 2)J (k ) = 0,where 1. Rearranging yields (d2d 2+1 dd 2 2)J (k ) =k2J (k ).ThusJ (k ) is an eigenfunction of the linear differential operatorL= (d2d 2+1 dd 2 2)with operatorLis not self-adjoint with respect to the standard scalar product, as the coefficientsp0( ) = 1 andp1( ) = 1/ do not satisfy the conditionp 1( ) =p0( ), so we use the weightfunctionw( ) =1p0( )e p1( )p0( )d = e 1/ d = eln =.

2 It follows from the relation( u v) bav (x)u(x)w(x)dx=(w(x)p0(x)(v (x)u (x) (v ) (x)u(x)) bathat a0 J (k )J (k )d =a[k J (ka)J (k a) kJ (k a)J (ka)]k2 k , in order to ensure Orthogonality , we must havekaandk abe zeros ofJ . Thus we havethe Orthogonality relation a0 J ( i a)J ( j a)d = 0, i6=j,where jis thejth zero ofJ .It is worth noting that because of the weight function being the Jacobian of the change ofvariable to polar coordinates, Bessel Functions that are scaled as in the above Orthogonality relationare also orthogonal with respect to theunweightedscalar product over a circle of that we haveorthogonalBessel Functions , we seekorthonormalBessel Functions . From a0 [J (k )]2d = limk ka[k J (ka)J (k a) kJ (k a)J (ka)]k2 k iand applying l Hospital s Rule yields a0 [J (k )]2d = limk k kaJ (k a)J ( i)k2 k 2=a22[J ( i)] the recurrence relationJ (x) = J 1(x) + 1xJ 1(x),we then obtain a0 [J ( ia )]2d =a22[J +1( i)] SeriesNow we can easily describe Functions as series of Bessel Functions .)

3 Iff( ) has the expansionf( ) = j=1c jJ ( j a),0 a, > 1,Then, the coefficientsc jare given byc j= J ( j a)|f( ) J ( j a)|J ( j a) =2a2[J +1( j)]2 a0 J ( j a)f( )d .It is worth noting that orthonormal sets of Bessel Functions can also be obtained by imposingNeumann boundary conditionsJ (k ) = 0 at =a, in which caseka= j, where jis thejthzero ofJ .We now consider an example in which a Bessel series is used to describe a solution of a Laplace s equation in a hollow cylinder of radiusawith endcaps atz= 0 andz=h, 2V= 0,with boundary conditionsV(a, ,z) = 0, V( , ,0) = 0, V( , ,h) =f( , )for a given potential functionf( , ). In cylindrical coordinates, Laplace s equation becomes1 ( V )+1 2 2V 2+ 2V z2= separation of variables, we assume a solution of the formV( , ,z) =P( ) ( )Z(z).

4 2 Substituting this form into the PDE and dividing byVyields1 Pdd ( dPd )+1 2 d2 d 2+1Zd2 Zdz2= to periodicity, we require that satisfyd2 d 2= m2 ,wheremis an integer, and therefore ( ) =eim . We also haved2 Zdz2=`2Z,which has solutionse`z. Because of the boundary condition atz= 0, we take linear combinationsof solutions so thatZ(z) = , we have 2d2Pd 2+ dPd + (`2 2 m2)P= is the Bessel equation of orderm, which has solutionsJm(` ). To satisfy the boundarycondition at =a, we set`= mj/a, where mjis thejth zero ofJm. We then haveV( , ,z) =Jm( mj a)eim sinh( mjza).To satisfy the boundary condition atz=h, we take a linear combination of solutions of thisand seek the coefficients of the expansionf( , ) = m= j=1cmjJm( mj a)eim sinh( mjha).

5 From the Orthogonality relation for Bessel Functions , as well as the Orthogonality relation 2 0e im eim d = 2 mm ,we obtaincmj=1 a2sinh( mjha)J2m+1( mj) 2 0 a0 e im Jm( mj a)f( , )d d .Thus the PDE and all boundary conditions are