My problem can be easily solved without Mathematica, however, I am curious how to do it in Mathematica. Moreover, I can be stuck with a similar, but more cumbersome problems in the future.
Following this topic, I defined the two complex differential operators of interest:
b1[r_, ph_] = Exp[I ph] (D[#, r] + I/r D[#, ph]) & b2[r_, ph_] = Exp[-I ph] (D[#, r] - I/r D[#, ph]) &
Now I can easily apply those to any functions of interest:
Now I want to define an operator that is a product (more precisely, superposition) of the two operators above. Note, that in my case the superposition is Laplacian operator in polar coordinates.
I've tried to do:
b3[r_, ph_] = b1[r, ph][b2[r, ph] [&]]
And some similar things. Can anyone help me to figure out how to define operators' superposition (so that
b3[r, ph][r^3] would return me the expression of interest)?