I often encounter the following statements:
$${D \over e^D - 1} = {\log(\Delta + 1) \over \Delta}$$
$$\int_x^{x+1} f(t)\,dt= {e^D - 1 \over D} [f]$$
$$\Delta = (e^D - 1)\,$$
$$f(a+x)=e^{a D}[f]$$
$$f(a x)=a^{x D}[f]$$
$$f\left(\frac x{1-x}\right)= e^{x^2 D}[f]$$
and so on. Where can I find
- the complete set of the rules of such manipulations
- whether the manipulations are applicable to non-linear operators
- the list of operators in this form (say, convolution operator, integration operator, composition etc)
- Whether the application of such construct to a function distributive (that is whether ${e^D - 1 \over D} f={e^{Df} - 1 \over Df}$
Any other info is also appreciated.