Solution of a Modified Bessel Differential Equation with Complex Coefficient

Consider the following ordinary differential equation (ODE)

$$\Big[\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr}-\big(r_i+\frac{\mu_m}{r^2}\big)\Big]R(r)=0$$

where $r$ is a real variable, $\mu_m$ is a real constant and $r_i$ is a complex constant. I already know that if $r_i$ is a negative or positive real number then the solution of this ODE is given by

\begin{align*}
R(r)&=C_1 I_{\mu_m}(\sqrt{r_i}\,r)+C_2K_{\mu_m}(\sqrt{r_i}\,r),\qquad r_i\gt0 \\
R(r)&=C_1 J_{\mu_m}(\sqrt{r_i}\,r)+C_2Y_{\mu_m}(\sqrt{r_i}\,r),\qquad r_i\lt0
\end{align*}

where $C_1,\,C_2$ are real constants and $I_{\mu_m}$ and $K_{\mu_m}$ are modified Bessel functions of order $\mu_m$. Also, $J_{\mu_m}$ and $Y_{\mu_m}$ are the Bessel functions of order $\mu_m$.

What will be the solution when $r_i$ is a complex constant?

Answer

For a purely imaginary $r_i$, the solutions are linear combinations of Kelvin functions. Thus, for a general complex $r_i$, solutions are superposition of Bessel or modified Bessel functions (depending on the real part of the parameter) and Kelvin functions.