Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

233

This equation is known as the index equation of the differential equation (5.38). The remaining coefficients a k can now be determined using λ and a 0 , and the formal solution of the initial equation can be written in the following form

∞ ∑ k = 1

y ( x , λ )= a 0 x λ

λ

k .

a k ( λ ) x

+ x

(5.41)

However, the solution is formal, as the convergence of this series has not been proved. The proof can be found in various books from this area. See, for example, [62].

Theorem15 Let us observe the differential equation in the following form x 2 y ′′ + xb ( x ) y ′ + c ( x ) y = 0 ,

(5.42)

and assume that b ( x ) and c ( x ) are analytical functions at point x = 0. If these functions can be replaced by series that are convergent for | x | < R , and λ i ( Re λ 1 > Re λ 2 , i = 1 , 2 ) are the solutions of the index equation λ ( λ − 1 )+ b ( 0 ) λ + c ( 0 )= 0 , (5.43) then a) differential equation (5.42) has two linearly independent solutions

∞ ∑ k = 0

a ( i )

( i ) 0 = 1 , i = 1 , 2 ,

λ i

k , a

y i ( x )= | x |

k x

(5.44)

if λ 1 and λ 2 are not equal, and their difference λ 1 − λ 2 is not a positive integer. The respective series ( y i ) are convergent for 0 < | x | < R . b) Differential equation (5.42) has two solutions in the format

∞ ∑ k = 0

a ( 1 )

λ 1

λ 1 r

k = | x |

y 1 ( x )= | x |

k x

1 ( x ) ,

∞ ∑ k = 0

a ( 2 )

λ 1 + 1

λ 1 + 1 r

k + cy

y 2 ( x )= | x | 2 ( x )+ cy 1 ( x ) · log | x | , if λ 1 = λ 2 . The respective power series r 1 ( x ) and r 2 ( x ) are convergent for 0 < | x | < R and r 1 ( 0 )̸= 0. c) Differential equation (5.42) has two solutions in the format y 1 ( x )= | x | λ 1 q 1 ( x ) , y 2 ( x )= | x | λ 2 q 2 ( x )+ cy 1 ( x ) · log | x | = | x | λ 1 r 2 ( x )+ cy 1 ( x ) · log | x | , if the difference λ 1 − λ 2 is a positive integer. The power series q 1 ( x ) and q 2 ( x ) are convergent for 0 < | x | < R and q i ( 0 )̸= 0. c is a constant, which may also be equal to zero. k x 1 ( x ) · log | x | = | x |