BESSEL FUNCTIONS
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[CHAP. 10
FUNCTIONS RELATED TO BESSEL FUNCTIONS 1. Hankel Functions of First and Second Kinds are defined respectively by 2. Modified Bessel Functions. The modified Bessel function of the first kind of order n is defined as (14) If n is an integer, but if n is not an integer, ln(x) and I-n(x) are linearly independent.
(15)
The modified Bessel function of the second kind of order n is defined as (16)
These functions satisfy the differential equation (17)
and the general solution of this equation is (18)
or if n + 0,1,2,3, ...
(19)
3. Ber, Bei, Ker, Kei Functions. The functions Bern(a;) and Bein(a;) are the real and imaginary parts of Jn(is/2x) where t372 = e3lri/4 = (\/2/2)(l — i), i.e. Jn(i3/2x) = Eern(x)+iEein(x) (20) The functions Kern(a;) and Kein(a;) are the real and imaginary parts of e~nMZKn(iV2x) where i1'2 = e™" = (v^/2)(l + t), i.e. (21) The functions are useful in connection with the equation (22)
which arises in electrical engineering and other fields. The general solution of this equation is (23) EQUATIONS TRANSFORMED INTO BESSEL'S EQUATION
The equation (24)
where k, a, r, /? are constants, has the general solution where [seepage 76].
(**) If « = 0 the equation is solvable as an Euler or Cauchy equation