Bessel’s equation

The series converge for all x. Much of the importance of Bessel s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. 

Bessel functions are prominent in theoretical chemistry and physics. These functions were first obtained as solutions of Bessel s equation ... 

Bessel s equation. What we hnvc shown js Hint if the n which occurs in Bessel s equation is an integer, one solution of the equation is J (ar). It is because of this fnct that Bessel coefficients tire of such importance in mathematical physics, for ns we saw in 1, the equation (20.0) arises naturally in boundary value problems in mathematical physics. 

Bessel s Differential Equation. We showed previously 2fi above) that, if n is an integer, J (ie) is a solution of Bessel s equation (2G. )). We shall now examine the solutions of that equation when the parameter n is not necessarily an integer. To emphasise that this parameter is, in general, non-integral, wc shall replace it by the symbol v, so that wc now consider the solutions of the second order linear differential equation... 

Show that the complete solution of Bessel s equation may be written in the form... 

The differential equation is the same for -n as for n, so D. is also a solution, and is generally different from D . Thus, a general solution of Bessel s equation with two arbitrary constants is... 

However, when n is integral, from the recurrence relations D = (-l)"Dn. This implies that D n is linearly dependent of Dn, and a second linearly independent solution to Bessel s equation must still be found. 

This function is called the Bessel function of the first kind of order n. T(n +1) is the gamma function of n +1. From this, it can be seen that when n is a positive integer, J (x) starts off as x". When n = 0, Jo(0) = 1. When n is an integer, J (0) = 0. In all other cases, J is infinite at the origin. In many physical problems the solution to Bessel s equation must be defined (finite) and well-behaved at the origin, which eliminates all solutions except for those with integer values of n. It can also be shown that J satisfies the same recurrence relations as Dn, verifying that the functions are the same. 

Bessel s equation. Linear differential equation xly" + xy + x2 - tt2)y = 0, whose solutions are expressible as power series in x. 

In the present case, in which the basic governing equation is linear, the asymptotic analysis serves only to simplify the solution procedure, for example, by avoiding the need to deal with Bessel s equation when Rn> 1. Later, however, we shall see that the same basic methods may often allow approximate analytic solutions to be obtained for nonlinear problems, even when no exact solution is possible. 

The homogeneous equation is a slightly disguised form of the modified Bessel s equation of order 0. 

The additional properties of these functions may all be derived from the above relations and the known properties of ] x). 12. Complete solutions to Bessel s equation may be written as... 

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