I'm looking at the $\beta-\gamma$ ghost CFT, although my question holds more generally for any OPE in any conformal field theory.
The OPE of the two defining fields is $$\beta(z) \gamma(w) \sim \frac{1}{z-w}.$$
To do calculations I also need to know the OPE $$\gamma(z) \beta(w)\sim \;?\;.$$ How can I calculate this using the first OPE? I know this OPE needs to be $\pm$ the first one (and I also know that won't hold more generally for any OPE).
Is the point that I can swap the operators around however I like because they're radially ordered, and so $\beta(z) \gamma(w) = \gamma(w)\beta(z)$?
I tried to repurpose this answer to my current question, but I've always found the OPE a bit cryptic and I need it spelt out to me in simple baby steps.