Integral symbol for closed loops over functions?

[jackmell]

I find it sometimes confusing dealing with integrals of multi-valued functions in distinguishing a closed integration path, and an integration path which forms a closed loop over the function. They can of course be quite different. For example:

$$\oint f(z)dz.$$

Now, is the integration to be done simply over any of the multiple sheets, for example random every $\pi/12$ over a circle or, is the integration to be done over an analytically-continuous path over the function?

Why don't we have a special integral symbol for the later?

 

[SteamKing]

Usually, the particular path of the integral will be described somewhere, either in text accompanying the integral or with some mathematical notation. Not all path integrals are necessarily evaluated over a circular, or even a curved, path.