ABSTRACT
CHAPTER 1
1. The Bessel differential equation. Application of power series. Cylindrical functions of the first kind
Consider the Bessel differential equation with the index v
z2 d2u dz2 + z
du dz +
(z2 – v2)u = 0. (1.1)
Equation (1.1) is a linear differential equation of second order; hence, its general integral can be expressed in the form
u(z) = C1u1(z) + C2u2(z),
where u1(z) and u2(z) are linearly independent partial solutions of equation (1.1). Suppose that z and v can admit any complex values. In the cases when the index v is an integer, we shall denote it by the letter n. If the argument z is a real number, then we shall denote it by the letter x. We introduce the Bessel operator of index v
∇v ≡ z2 d2
dz2 + z d dz
+ z2 – v2 (1.2)
and we rewrite (1.1) in the following form: ∇v u = 0. We shall seek a solution of equation (1.1) in the form of a generalized power
series in increasing powers of the argument z
u(z) = ∞
αm zm+α , (1.3)
where α0 ≠ 0. Let us determine α and the coefficients αm of series (1.3) here. For this purpose
we find the first and the second derivatives of (1.3)
u'(z) = ∞
m=0 α m ( m + α)zm+α –1
u"(z) = ∞
m=0 α m ( m + α)(m + α – l)zm+α – 2
and substitute the series obtained into the left-hand side of equation (1.1) instead of the function u(z).
