Constant coefficients

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In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of operators having variable coefficients. Such constant coefficient operators have been found to be the easiest to handle, in several respects. They include for example the Laplacian of potential theory and other major examples of mathematical physics.

In the case of ordinary differential equations, writing

D = d/dx

the general constant-coefficient differential operator is

L = p(D),

where p is any polynomial with complex number coefficients. The solution of equations

Lf = g

with a given function g(x) was given already in the eighteenth century, by Leonhard Euler.

For partial differential equations, the constant-coefficient operators are characterised geometrically by their translation invariance, and algebraically as polynomials in the partial derivatives. According to the Ehrenpreis–Malgrange theorem, they all have fundamental solutions.