Analytization trick

The analytization trick is a heuristic often applied by physicists.

Suppose we have a function f of a complex variable z which is not analytic, but happens to be differentiable with respect to its real and imaginary components separately. Differentiating f with respect to z is out of the question, but it turns out if

${\displaystyle f(z)=g({\bar {z}},z)}$

for some analytic function g of two complex variables, we can pretend f is g (physicists do this sort of thing all the time) and work with

${\displaystyle \left.{\frac {\partial }{\partial z_{1}}}g\right|_{z_{1}={\bar {z}};z_{2}=z}}$

and

${\displaystyle \left.{\frac {\partial }{\partial z_{2}}}g\right|_{z_{1}={\bar {z}};z_{2}=z}}$

instead. Physicists write these as

${\displaystyle {\frac {\partial }{\partial {\bar {z}}}}f({\bar {z}},z)}$

and

${\displaystyle {\frac {\partial }{\partial z}}f({\bar {z}},z)}$

and give some handwaving explanation as to why ${\displaystyle {\bar {z}}}$ and z may be treated as if they are "independent" when they really are not.

Note that if g exists, it is unique (due to the theorem about the uniqueness of analytic continuations), at least if we ignore complications like branch cuts and so on.

Conceptually, whenever this trick is used, it probably means on a physical level that the variable z that they are working with "really" has a real structure and physicists are merely pigeonholing it into a complex variable.

Actually, it's not even necessary for there to be an analytic g. It's enough for f to be differentiable with respect to its real and imaginary components (or n times differentiable, as the case may be). In that case,

${\displaystyle f({\bar {z}},z)}$

has to be treated purely formally.