# What is operator calculus?

I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.

I have searched for "operator calculus" and have not found any accessible references that introduce the topic. Maybe operator calculus go under another name today, than at the time of the interview?

Can you give me a reasonably simple explanation of operator calculus, and also give some references to books on the subject?

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The tag "calculus" is not really appropriate. I think functional calculus is another word for it, you might find something if you look for that. –  Jonas Teuwen Nov 13 '10 at 19:47
I believe he was referring to fractional calculus. Check out the wikipedia page: en.wikipedia.org/wiki/Fractional_calculus and if there is a specific point you find confusing, you may want to edit your question to address that specific point. Hope this helps :) –  WWright Nov 13 '10 at 19:56
The following article might help to determine what he was talking about: jstor.org/pss/2028275 –  Adrián Barquero Nov 13 '10 at 20:01

At the heart the key idea is quite simple. Namely, one views functions, endomorphisms, etc. as "numbers" and manipulates them as such (where valid). E.g. for a derivative $\rm D = d/dx\$ we have that $\rm D^2 - c^2\ =\ (D - c)(D + c)$ just as for numbers, as long $\rm\:D\:$ commutes with $\rm\:c\:$, i.e. $\rm\:c\:$ is constant. So we can solve constant coefficient linear differential / difference equations by simply factoring their operator ("characteristic") polynomials into linear factors over $\mathbb C$. One can also similarly perform transcendental operator manipulations such as

$$\rm f(D)\ e^{t\:x}\ =\ f(t)\ e^{t\ x}$$

$$\rm e^{\:t\ D}\ f(x)\ =\ f(x+t)$$

$$\rm t^{x\ D}\ f(x)\ =\ f(t\: x)$$

and the Generalized Leibniz Rule

$$\rm g(D)\ f(x)\ = \ \sum_{n=0}^\infty\ \frac{f^{(n)}(x)\ g^{(n)}(D)}{n!}$$

For further details see Roman: Umbral Calculus and Rainville: Special Functions and especially Rota: Finite operator calculus.

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Also see Erwin Kreyszig, Advanced Engineering Mathematics, for worked out examples. I should also mention that the method was created by the electrical engineer Oliver Heaviside for solving problems of electrical and control networks, and a rigorous justification of such methods can be found in the theory of integral transforms, for instance Laplace transforms. And for some of the things such as Dirac Delta function and its derivatives, you need the theory of distributions as well. –  user1119 Nov 13 '10 at 20:23