Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

231

Figure 5.1: Legendre polynomials.

Starting from the binomial formula, applied to ( x 2 − 1 ) n , and differentiating this expression n times, term by term, we obtain the so called Rodrigues 3 formula

d k d x k h

k i

1 2 k k !

x 2 − 1

P k ( x )=

(5.31)

.

An example of application of Legendre polynomials in geophysics is geomagnetic potential [72]. Namely, for calculating the magnetic potential at the Earth’s surface, a function is used of the form U = R ∞ ∑ n = 1 ∞ ∑ m = 0 [ g m o cos ( m λ )+ h m o sin ( m λ )] P m n [ cos Θ ] , where R is Earth’s radius, g m o and h m o coefficients that depend on the basic characteristics of the magnetic field, and P m n are Legendre polynomials (see Example 197, p. 282).

5.4 Bessel equation. Bessel functions

Let us first introduce some concepts and theorems that are going to be used when solving Bessel equation. Observe the linear differential equation of n thorder

( n ) + ··· + a

x ) y ′ + a n ( x ) y = 0 .

a 0 ( x ) y

(5.32)

n − 1 (

Definition The point x 0 , at which the condition a 0 ( x 0 )= 0 is fulfilled, is called a singular point .

3 Olinde Rodrigues (1794-1851), French mathematician and economist.