I wonder, what are the meanings of elementary functions applied to linear operators as matrices. Particularly, I was interested of logarithm of derivative operator (if it exists).
For instance, when a function $\phi(x)$ applied to the derivative operator, via Fourier transform,
$$\phi [D] f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\phi(-i\omega) \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$
This way,
$e^D-1=\Delta=f(x+1)-f(x)$
$\frac1{D}=D^{-1}=\int f(x)dx$
etc.
But what about logarithm? I also found matrices for sine, cosine and tangent of derivative operator, but do they have any useful meaning?