I originally saw the technique in the accepted answer of this Stack Exchange post: Fastest way to solve a linear system of ODEs.
Consider a system of coupled linear ODEs:
$$ x'(t) = -k y(t) $$ $$ y'(t) = k x(t) $$
The solution technique defines a new variable, $z=x+iy$ where $i=\sqrt{-1}$. The two equations can be combined as $i$ times the first plus the second.
$$ -ix'(t) + y'(t) = k(iy(t) + x(t)) \rightarrow z'(t) = ikz(t) $$
The single ODE for $z$ can then be solved and the solutions taken as
$$ x(t) = \Re(z(t)), $$ $$ y(t) = \Im(z(t)). $$
What is the name of this technique? I'm looking for some textbooks and websites that discuss it.
I've seen a related approach in the physics textbook ElectroMagnetism by Polluck and Stump. They refer to the procedure as using Conjugate Functions.
Suppose you have complex function $f(z)=u(x,y)+iv(x,y)$. The funciton is Analytic iff $u_x=v_y$ and $u_y=-v_x$ where the subscript denotes a partial derivative.
Note these equations imply $u_{xx}+u_{yy}=v_{xx}+v_{yy}=0$.
From P&S:
" What is the potential function V (x, y) for a region bounded by charged conducting plates that intersect at a right angle?
We will specify $V = \Phi_0$ on the boundary, but the precise value of <1>0 is not particularly relevant because we can always add a constant to the potential.
$F(z) = Az^2 + i\Phi_o = A(x^2 - y^2) + 2iAxy + i\Phi_o, $ "
This technique is frequently used when the geometry is roughly 2 dimensional, e.g. two planes intersecting at an arbitrary angle. Consdiered two 2 because only the 2d cross section is relevant.