ned to think that the Fourier method would be most ideal as it is the de facto choice among physicists studying extension theory in General relativity.

There is also the Michell solution for the Biharmonic equation. (en.wikipedia.org/wiki/Michell_solution) It really just boils down to making a variable substitution $r \mapsto e^{t}$ and factoring a quartic polynomial. If I recall correctly, IIT has lecture notes available for the derivation. Unfortunately, $B\Delta u$ and $Cu$ will ruin the necessary symmetry for the change of variables. That being said, you could still produce a recurrence relationship that could probably be solved numerically. Of what theoretical value such a formula would be is outside of my knowledge. I am incli

Thank you for your excellent answer. I'll try to study the material and respond with further questions on the chat. When you say that the Fourier transform of $g$ allows complex arguments, does $g$ admit a full Fourier transform of the form $f(x) = \sum_{n=-\infty}^{\infty} A_{n} e^{-inx}$ where $A_n$ is calculated from $g$? Thanks again.

Thank you for your comment, I'll work to revise my question, and will remove the extraneous statement about an inner product. Do you have an idea about separating the real and imaginary components of the Schwarz-Christoffel Transform? Would there be an alternative method to calculate its Fourier transform. Thank you again.