Linear Translation in Polar Coordinates

All across literature and the web one can find formulae, e.g. for converting cartesian to polar coodrinates or for translation and rotation in cartesian coordinates.

Why bother?

But recently I ran into the problem of requiring a two dimensional integral in polar coordinates due to some radial symmetry. Unfortunately within this symmetry I have another but linearly displaced radial symmetry. In order to perform the integral I had to find a formuale for describing the linear translation in polar coordinates.


Nowhere in the web I found a formula. So I had to derive an expression from the linear translation vector in cartesion coordinates and convert that one to polar coordinates.