Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

237

Figure 5.2: Bessel functions of the first kind.

Figure 5.3: Bessel functions of the second kind.

Definition The function defined by the relation

∞ ∑ s = 1

1 s

( − 1 ) s ( s ! ) 2

x 2

1 2

2 s

K 0 = J 0 log x −

1 +

(5.63)

+ ··· +

is called Bessel function of the second kind of zero order .

This function i known in literature also as Neumann 7 or Macdonald function of zero order. Let us finally summarize. For ν = 0, Bessel equation (5.45) has two linearly independent solutions y 1 = J 0 , y 2 = K 0 , (5.64) that is, in this case the solution of the equation is where a and b are arbitrary constants. When solving the index equation that corresponds to Bessel equation (5.45) we have assumed that ν = 0. Now consider the case when ν̸ = 0. In this case, λ 1̸ = λ 2 and λ 1 − λ 2 = 2 ν . (5.66) As we have assumed that ν is a real parameter, we will observe two cases, when the difference λ 1̸ = λ 2 is an integer, and when it is not an integer. Consider the case when 2 ν is not integer. According to Theorem 15 a), in this case there are two linearly independent solutions of the form y = aJ 0 + bK 0 , (5.65)

∞ ∑ k = 0

a ( i )

λ i

k , i = 1 , 2 .

y i = | x |

k x

(5.67)

7 Carl Neumann (1832-1925), German mathematician and physicist.