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Concept
Wronskian
Summary
In the mathematics of a square matrix, the Wronskian (or Wrońskian) is a determinant introduced by the Polish mathematician . It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions.
The Wronskian of two differentiable functions f and g is .
More generally, for n real- or complex-valued functions f1, ..., fn, which are n – 1 times differentiable on an interval I, the Wronskian is a function on defined by
This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the derivative, thus forming a square matrix.
When the functions fi are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly. (See below.)
If the functions fi are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.
A common misconception is that W = 0 everywhere implies linear dependence, but pointed out that the functions x2 and x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0. There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.
Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.
gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n – 1 of them do not all vanish at any point then the functions are linearly dependent.
Official source
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