Operational calculus

Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.
History.
The idea of representing the processes of calculus, differentiation and integration, as operators 
has a long history that goes back to Gottfried Wilhelm Leibniz. The mathematician Louis François Antoine Arbogast was one of the first to manipulate these symbols independently of the function to which they were applied.
This approach was further developed by Francois-Joseph Servois who developed convenient notations. Servois was followed by a school of British and Irish mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester.
Treatises describing the application of operator methods to ordinary and partial differential equations were written by Robert Bell Carmichael in 1855 and by Boole in 1859.
This technique was fully developed by the physicist Oliver Heaviside in 1893, in connection with his work in telegraphy. 
At the time, Heaviside's methods were not rigorous, and his work was not further developed by mathematicians. 
Operational calculus first found applications in electrical engineering problems, for 
the calculation of transients in linear circuits after 1910, under the impulse of Ernst Julius Berg, John Renshaw Carson and Vannevar Bush.
A rigorous mathematical justification of Heaviside's operational methods came only 
after the work of Bromwich that related operational calculus with 
Laplace transformation methods (see the books by Jeffreys, by Carslaw or by MacLachlan for a detailed exposition). 
Other ways of justifying the operational methods of Heaviside were introduced in the mid-1920s using 
integral equation techniques (as done by Carson) or Fourier transformation (as done by Norbert Wiener).
A different approach to operational calculus was developed in the 1930s by Polish mathematician 
Jan Mikusiński, using algebraic reasoning.
Norbert Wiener laid the foundations for operator theory in his review of the existential status of the operational calculus in 1926:
Principle.
The key element of the operational calculus is to consider differentiation as an operator p = acting on functions. Linear differential equations can then be recast in the form of "functions" of the operator p acting on the unknown function equaling the known function. Here, is defining something that takes in an operator p and returns another operator . 
Solutions are then obtained by making the inverse operator of act on the known function. The operational calculus generally is typified by two symbols, the operator p, and the unit function 1. The operator in its use probably is more mathematical than physical, the unit function more physical than mathematical. The operator p in the Heaviside calculus initially is to represent the time differentiator . Further, it is desired this operator bear the reciprocal relation such that p denotes the operation of integration.
In electrical circuit theory, one is trying to determine the response of an electrical circuit to an impulse. Due to linearity, it is enough to consider a unit step:
The simplest example of application of the operational calculus is to solve: , which gives
From this example, one sees that formula_2 represents integration. Furthermore iterated integrations is represented by formula_3 so that 
Continuing to treat p as if it were a variable,
which can be rewritten by using a geometric series expansion, 
Using partial fraction decomposition, one can define any fraction in the operator p and compute its action on . 
Moreover, if the function 1/"F"(p) has a series expansion of the form
it is straightforward to find
Applying this rule, solving any linear differential equation is reduced to a purely algebraic problem.
Heaviside went further, and defined fractional power of p, thus establishing a connection between operational calculus and fractional calculus. 
Using the Taylor expansion, one can also verify the Lagrange-Boole translation formula, , so the operational
calculus is also applicable to finite difference equations and to electrical engineering problems with delayed signals.