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A ROS API Documentation per-se doesn't exist.
ROS is composed of many many different modules (I suggest you check the original PDF) each of which doing a very small part of the work.
The nearest thing to a baseline ROS API is roscpp which is the foundation of ROS. In this page (as on most other pages) on the top right you'll find a link Code API which will take you to the relevant doxygen.
Good luck learning ROS!