Transcription of Orthogonality of Bessel Functions - USM
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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 =.
Orthogonality of Bessel Functions Since Bessel functions often appear in solutions of PDE, it is necessary to be able to compute coe cients of series whose terms include Bessel functions. Therefore, we need to understand their orthogonality properties. Consider the Bessel equation ˆ2 d2J (kˆ) dˆ2 + ˆ dJ (kˆ) dˆ + (k2ˆ2 2)J (kˆ) = 0 ...
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