Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONS10.5Y 0Y 1 Y22 4 6 8 10 x−0.5−1Figure 18.6The first three integer-order Bessel functions of the second kind.18.5.3 Properties of Bessel functions J ν (x)In physical applications, we often require that the solution is regular at x =0,but, from its definition (18.81) or (18.82), it is clear that Y ν (x) is singular atthe origin, and so in such physical situations the coefficient c 2 in (18.83) mustbe set to zero; the solution is then simply some multiple of J ν (x). These Besselfunctions of the first kind have various useful properties that are worthy offurther discussion. Unless otherwise stated, the results presented in this sectionapply to Bessel functions J ν (x) of integer and non-integer order.Mutual orthogonalityIn section 17.4, we noted that Bessel’s equation (18.73) could be put into conventionalSturm–Liouville form with p = x, q = −ν 2 /x, λ = α 2 and ρ = x,provided αx is the argument of y. From the form of p, we see that there is nonatural interval over which one would expect the solutions of Bessel’s equationcorresponding to different eigenvalues λ (but fixed ν) to be automatically orthogonal.Nevertheless, provided the Bessel functions satisfied appropriate boundaryconditions, we would expect them to obey an orthogonality relationship oversome interval [a, b] oftheform∫ baxJ ν (αx)J ν (βx) dx =0 forα ≠ β. (18.84)608