SPECIAL FUNCTIONS
Y 0
Y (^1) Y
2
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x− 1
− 0. 5
0. 5
1
Figure 18.6 The first three integer-order Bessel functions of the second kind.18.5.3 Properties of Bessel functionsJν(x)In physical applications, we often require that the solution is regular atx=0,
but, from its definition (18.81) or (18.82), it is clear thatYν(x) is singular at
the origin, and so in such physical situations the coefficientc 2 in (18.83) must
be set to zero; the solution is then simply some multiple ofJν(x). These Bessel
functions of the first kind have various useful properties that are worthy of
further discussion. Unless otherwise stated, the results presented in this section
apply to Bessel functionsJν(x) of integer and non-integer order.
Mutual orthogonalityIn section 17.4, we noted that Bessel’s equation (18.73) could be put into con-
ventional Sturm–Liouville form withp=x,q=−ν^2 /x,λ=α^2 andρ=x,
providedαxis the argument ofy. From the form ofp, we see that there is no
natural interval over which one would expect the solutions of Bessel’s equation
corresponding to different eigenvaluesλ(but fixedν) to be automatically orthog-
onal. Nevertheless, provided the Bessel functions satisfied appropriate boundary
conditions, we would expect them to obey an orthogonality relationship over
some interval [a, b]oftheform
∫baxJν(αx)Jν(βx)dx=0 forα=β. (18.84)