Consider the following two couples of functions:
- Bessel function of the first kind $J_{\nu}$ and Bessel function of the second kind $N_{\nu}$, also known as $Y_{\nu}$;
- modified Bessel function of the first kind $I_{\nu}$ and modified Bessel function of the second kind $K_{\nu}$.
The second couple is used for the general solution of the modified Bessel (homogenous) differential equation:
$$f(x) = A I_{\nu}(x) + B K_{\nu} (x) \label{a} \tag{1}$$
But the same general solution can still be written in terms of the first couple as
$$g(x) = C J_{\nu}(ix) + D N_{\nu} (ix) \label{b} \tag{2}$$
where $i$ is the imaginary unit, according to the definition itself of the modified Bessel differential equation.
Which coefficients $C, D$ must be given to the functions in $\ref{b}$ in order to obtain the solution $\ref{a}$?
$C = i^{-\nu}$ immediately leads to $I_{\nu}(x)$, but I can not figure out how to proceed about the remaining part, not knowing a direct relationship between $K_{\nu}$ and $N_{\nu}$.
From an alternative perspective:
The linearly independent functions $I_{\nu}(x)$ and $K_{\nu}(x)$ generate the space of solutions of a modified Bessel differential equation. But also the linearly independent functions $J_{\nu}(ix)$ and $N_{\nu}(ix)$ generate the same space of solutions.
Can, then, the couple of functions $I_{\nu}(x)$, $K_{\nu}(x)$ be expressed in terms of $J_{\nu}(ix)$, $N_{\nu}(ix)$? If not, why?
If this were linear algebra, the answer would be yes. What instead about functional analysis?
Best Answer
You might check this Wikipedia page: https://en.wikipedia.org/wiki/Bessel_function