Where to find reference about dealing with operators in form of matrices?

Question

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.

Answer 1

As an intro to the operational calculus, you might try looking at Operational Calculus: Based on the two-sided Laplace Transform by Van der Pol and Bremmer and Lectures on Applications-Oriented Mathematics by Friedman. Also, a paper by Lindell, "Heaviside Operational Rules Applicable to Electromagnetic Problems," has a good overview with rules of action of many pseudo-differential operators (ref. from Dead Reckonings website).

Friedman's book gives you a taste of the applications, and many of the formulas you list are covered there, but it is not as rigorous or as systematic as Van der Pol and Bremmer's. These require somewhat heavy study with a good foundation in basic complex analysis.

The first book in the intro explains with a particularly simple operator the necessity of some systematic, consistent method of interpretation:

Which expression is correct,

A: $\displaystyle\frac {1}{1-D} = 1+D+D^2+D^3+\cdots$ or

B: $\displaystyle\frac {1}{1-D} = -\left(\frac {1}{D}+\frac{1}{D^2}+\frac{1}{D^3}+\cdots\right)$ with $\displaystyle\frac{1}{D}H(x) f(x) = H(x)\int^x_0 f(u) \, du$ ?

(H(x) is the Heaviside step function.)

PS: "The series is divergent, therefore we may be able to do something with it." - Heaviside.

Rigor aside, Heaviside often used operator expansions similar to that of A to generate an asymptotic series expansion for functions represented by convergent series generated by expansions similar to B . See Heaviside's Operator Calculus at Dead Reckonings and "The asymptotic solution of an operational equation" by Carson.

Also look at the finite operator calculus, or umbral calculus, associated with Blissard, Bell, Stephen Roman, and Gian-Carlo Rota, among others. Survey articles are available on the Net, e.g., An Introduction to Umbral Calculus by Di Bucchianico, with extensive bibliographies.

Additional references for miscellaneous differential ops and their actions:

H.T. Davis, The Theory of Linear Operators (e.g., p. 89)

K. Jordan, Calculus of Finite Differences

MathOverFlow: MO-107159, Pochhammer symbol of a differential MO-102281, A mysterious Heisenberg algebra identity from Sylvester

MathStackExchange: MSE-116633, MSE-126984, and MSE-169072

OEIS: OEIS-A145271, OEIS-A132440, OEIS-A132681, OEIS-AA094638, OEIS-A021009, OEIS-A218234 (Follow ref. for P. Blasiak and P. Flajolet, G. Dattoli, and W. Lang. Cf. also Merida Lectures--Lie Algebras, Representations, and Semigroups Through Dual Vector Fields by Philip Feinsilver.)

Answer 2

It's better to think of $D$ is being similar to a matrix, in that you can't really define division by $D$, and it is a singular (non-invertible) linear function on some vector space. So you can't in general define $F(D)$ for any $F$ - for example, $\frac{1}{D}$ doesn't make sense, because $D$ is not invertible. (You can see that $D$ is not invertible because $D(f+c)=Df$ for any constant $c$.)

So you are stuck with the kinds of operations you can do with matrices. One of the things you can do with matrices is put them in power series.

For example, if $M$ is a matrix, $e^{M}$ makes sense, when defined using the power series for $e^z$, and it converges for all $M$. $\frac{e^M-1}{M}$ does not strictly make sense, when $M$ is not invertible, but if we define it via the power series for $\frac{e^z-1}{z}$, then it does make sense. So, in this sense, we can only really work with power series, rather than with more general functions.

Answer 3

Operational calculus is a ubiquitous tool in the physicist's or engineer's toolbox, and any of the excellent references and be rummaged for tricks and seat-of-the pants adaptations.

As your examples illustrate, the basic tool is the shift operator introduced by Lagrange, on the basis of the Taylor expansion.

To the extent the other fine answers did not remind you of it, most of the expressions you give are mere changes of variables to canonical coordinates for advection of that basic shift operator. You simply seek the suitable coordinates for the problem.

Thus, starting from $$e^{a\frac{d}{dx}} ~f(x)= f(a+x),$$ you appreciate that $$ \large a^{x\frac{d}{dx}} ~f(x)=e^{\ln a ~\frac{d}{d\ln x}} ~f(e^{\ln x})=f(e^{\ln x+\ln a})=f(a x)\small , $$ or else, for $y=-1/x$, $$\large e^{ax^2\! \frac{d}{dx}}~f(x)= e^{a\frac{d}{dy}}~f\left(-\frac{1}{y}\right)= f\left(-\frac{1}{y+a}\right)= f\left(\frac x{1-a x}\right)\small , $$ etc., illustrated in the WP article cited. The bulk of such applications are handled by conversion to canonical coordinates that are you to a translation operator.

This advection underlies the translation functional equation which, in turn, is at the foundation of Abelian Lie group theory.