## Bessel Functions Question

Hi, everybody.

I am trying to write a short article on Bessel functions but need a little help.

Bessel functions arise as the solutions to Bessel’s differential equation:

$x^2y'' + xy' + (x^2 - \alpha^2)y = 0$

where $\alpha$ is a constant.

My old calculus book includes an introductory chapter on differential equations which states that, because this is a second-order differential equation, it should have two linearly-independent solutions:

(i) $J\alpha(x)$ and (ii) $Y\alpha(x)$.

In addition, a linear combination of these solutions is also a solution:

(iii) $H\alpha(x) = C1 J\alpha(x) + C2 Y\alpha(x)$

where C1 and C2 are constants.

These are Bessel functions of the first, second, and third kind.

My question arises because other references, specific to Bessel’s functions, indicate that Bessel functions of the third kind actually take the form of two equations:

(iiia) $H\alpha^{(1)}(x) = J\alpha(x) + Y\alpha(x) i$
(iiib) $H\alpha^{(2)}(x) = J\alpha(x) - Y\alpha(x) i$

where i indicates the imaginary component (the square root of negative 1).

Anybody know how to go from (iii) to (iiia) and (iiib)? Is it something that can be easily explained here? Or perhaps you could direct me to a good reference that shows how the last two equations are derived?