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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?

Anixx
  • 9,119
  • One way of making sense of functions of operators is via power series. If there is a power series for a function, that's something you can attempt to plug an operator into, and get a corresponding function of an operator (there may be convergence issues). There are also definitions involving contour integration. The spectral theorem is a way of making this more precise. It is subtler for potentially unbounded operators like differentiation but it is still a thing, and very helpful. See e.g. https://math.stackexchange.com/questions/149842/understanding-the-symmetric-hermitian-matrix-operator – leslie townes Feb 01 '21 at 01:50
  • @leslietownes I wonder, what is logarithm of derivative? – Anixx Feb 01 '21 at 01:52
  • I don't have a well-developed intuition. If you have what is called a "functional calculus" for operators (i.e., a map that, given a function f and an operator A, gives you an operator f(A) in a way that respects addition and multiplication), functions of an operator are just whatever the definitions spit out. They may not be interesting or useful, in the same way that computing what $\sin(4.6525)$ is may not be more interesting or useful than knowing what $\sin(\pi)$ is. – leslie townes Feb 01 '21 at 01:56
  • @leslietownes Finding $\log(D+s)$ and plugging $s\to0$ gives infinities, but the same happens with $1/(D+s)$, but we still can find a matrix for integral. – Anixx Feb 01 '21 at 01:57

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