In mathematics, a Cauchy–Euler operator is a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
of the form
for a polynomial ''p''. It is named after
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
and
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
. The simplest example is that in which ''p''(''x'') = ''x'', which has eigenvalues ''n'' = 0, 1, 2, 3, ... and corresponding eigenfunctions ''x''
''n''.
See also
*
Cauchy–Euler equation
In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear differential equation, linear homogeneous differential equation, homogeneous ordinary differential equation with variable coefficients. ...
*
Sturm–Liouville theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form
\frac \left (x) \frac\right+ q(x)y = -\lambda w(x) y
for given functions p(x), q(x) and w(x), together with some ...
References
Differential operators
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