Mathematical Physics Vol 1

5.4 Bessel equation. Bessel functions

241

Now, according to (5.103), we can also calculate J 3 / 2 ( x )= r 2 π · x − cos x + sin x x = = r 2 π · x sin ( x − π / 2 )+ 1 x

cos ( x − π / 2 ) ,

(5.92)

J 5 / 2 ( x )= r

2 π · x − 2 π · x

3 x

1 x cos ( x − π / 2 ) =

sin ( x − π / 2 )+

sin x +

= r cos ( x − π ) . The last relation can be generalized for the purpose of calculating the Bessel function of the form J n + 1 / 2 J n + 1 / 2 ( x )= r 2 π · x P n 1 x sin x − n π 2 + Q n 1 x cos x − n π 2 (5.93) where P n and Q n are polynomials of 1 / x . From the last relation we can see that the Bessel function J n + 1 / 2 can be approximated by the expression J ν ( x )= r 2 π · x h cos x − νπ 2 − π 4 + O ( x − 1 ) i , x > 0 . (5.94) This asymptotic relation is valid not only for ν = n + 1 / 2, but also for every other ν . There are tables of values of Bessel functions at specific points. 1 − 3 x 2 sin ( x − π )+ 3 x Let us now find the general solution for Bessel equations. To that end we shall introduce a new function defined by Y ν ( x )= J ν ( x ) cos νπ − J − ν ( x ) sin νπ . (5.95) It can be proved that this function is a solution of the initial equation (5.45), as a linear combina tion of its solutions (the principle of superposition) when n is an integer. In that case the right hand side is an undetermined expression 0 / 0. From this expression, by using L’Hospitals rule, we obtain, for an integer ( ν = n )

5.4.2 Weber functions

k !

x 2

∞ ∑ k = 0

2 π

x 2 −

1 π

( n − k − 1 ) !

2 k − n

Y n ( x )=

J n ( x ) ln

(5.96)

−

( − 1 ) k

x 2 2 k − n k ! ( k + n ) !

Γ ′ ( n + k + 1 ) Γ ( n + k + 1 ) .

∞ ∑ k = 0

Γ ′ ( k + 1 ) Γ ( k + 1 )

1 π

−

+

9 function . The Weber function is a solution of Bessel

The function Y n is called the Weber

equation in the case when ν is an integer ( ν = n ). 9 Heinrich Weber (1842-1913), German mathematician.