In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have proven the following standard formula by changing the contour ($C_a$ is a contour around $a$): $$\int_{C_0} \frac{\mathrm{d}z}{2\pi i} [A(z),B(w)] = \int_{C_w} \frac{\mathrm{d}z}{2\pi i} R(A(z)B(w)).$$
This formula is obviously wrong. For example set $w = 0$. Then on the right side there is an integral over $A(z)B(0)$ and on the left side we have the same integral over $[A(z),B(0)]$ which is not the same (ok, maybe $w = 0$ is excluded).
But in general the right hand side is not well defined since the integral over $C_w$ is path dependent (since $R(A(z)B(w)$) has a branch cut at $|z| = |w|$). I checked this directly for $A(z) = \frac{A}{z}$ and $B(w) = B = const.$ (I can add the calculation if needed).
After all I think that I misunderstood radial ordering. We were never given an explicit definition. So how is radial ordering defined? And if my definition is wrong why can we use radial ordering in the same way as time ordering?
