Question

what is the formal definition of homogeneous medium?

Answer 1

just from the word homgeneous i would say it has uniform properties throughout

Answer 2

yes,i know but i want something more rigorous

Answer 3

You can say the medium has continuous translational and rotational symmetry, which means whether you move it or rotate it by any amount it looks the same. You could say all of the relevant properties are not functions of location. (If you want it to be homogenous in time you'll also need to specify they are not functions of time.)

Part of the difficulty with answering your question is that you haven't defined "formal" and the meaning of that word can be quite different, in different contexts.