How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or time harmonic. I am sure this problem has been dealt with before but where and what to look for. The current problem is in wave propagation and elastic waves. If an elastic wave propagates in a homogenous isotropic media has the following general solutions of the displacement potentials can be determined by applying Fourier expansion with respect to $\theta$ . The solution is $$ \varphi=\sum_{n=-\infty}^{\infty}\left\{ J_{n}\left(k_{L}r\right)\begin{array}{c} A_{n}^{(1)}\end{array}+H_{n}^{(1)}\left(k_{L}r\right)\begin{array}{c} A_{n}^{(2)}\end{array}\right\} e^{i\left(n\theta\right)}e^{j\omega t} $$ $$ \psi=\sum_{n=-\infty}^{\infty}\left\{ J_{n}\left(k_{s}r\right)\begin{array}{c} A_{n}^{(3)}\end{array}+H_{n}^{(1)}\left(k_{s}r\right)\begin{array}{c} A_{n}^{(4)}\end{array}\right\} e^{i\left(n\theta\right)}e^{j\omega t} $$
where $J_{n}\left(.\right)$ is Bessel function of the first kind and $H_{n}^{(1)}\left(.\right)$ is Hankel function of the first kind. $A_{n}$ potential displacement expanded amplitudes. The issue arises more in programming and dealing with complex numbers. Is it due to i or j?
