In mathematics, the Wrońskian of n differentiable functions is the determinant formed with the functions and their n – 1 first derivatives. It was introduced in 1812 by the Polish mathematician Józef Hoene-Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.

Definition

The Wrońskian 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 Wrońskian 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 Wrońskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly. (See below.)

The Wrońskian and linear independence

If the functions fi are linearly dependent, then so are the columns of the Wrońskian (since differentiation is a linear operation), and the Wrońskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wrońskian does not vanish identically. It may, however, vanish at isolated points.[1]

A common misconception is that W = 0 everywhere implies linear dependence. Peano (1889) pointed out that the functions x2 and |x| · x have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.[a] There are several extra conditions which combine with vanishing of the Wrońskian in an interval to imply linear dependence.

Over fields of positive characteristic p the Wrońskian may vanish even for linearly independent polynomials; for example, the Wrońskian of xp and 1 is identically 0.

Application to linear differential equations

In general, for an th order linear differential equation, if solutions are known, the last one can be determined by using the Wrońskian.

Consider the second order differential equation in Lagrange's notation:

where , are known, and y is the unknown function to be found. Let us call the two solutions of the equation and form their Wrońskian

Then differentiating and using the fact that obey the above differential equation shows that

Therefore, the Wrońskian obeys a simple first order differential equation and can be exactly solved:

where and is a constant.

Now suppose that we know one of the solutions, say . Then, by the definition of the Wrońskian, obeys a first order differential equation:

and can be solved exactly (at least in theory).

The method is easily generalized to higher order equations.

Generalized Wrońskians

For n functions of several variables, a generalized Wrońskian is a determinant of an n by n matrix with entries Di(fj) (with 0 ≤ i < n), where each Di is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the single variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson (1989b).

History

The Wrońskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII).

See also

Notes

  1. ^ Peano published his example twice, because the first time he published it, an editor, Paul Mansion, who had written a textbook incorrectly claiming that the vanishing of the Wrońskian implies linear dependence, added a footnote to Peano's paper claiming that this result is correct as long as neither function is identically zero. Peano's second paper pointed out that this footnote was nonsense.[2]

Citations

  1. ^ Bender, Carl M.; Orszag, Steven A. (1999) [1978], Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, New York: Springer, p. 9, ISBN 978-0-387-98931-0
  2. ^ Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. doi:10.4169/loci003642. Retrieved 2020-10-08.
  3. ^ Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. Section "On the Wronskian Determinant". doi:10.4169/loci003642. Retrieved 2020-10-08. The most famous theorem is attributed to Bocher, and states that if the Wronskian of analytic functions is zero, then the functions are linearly dependent ([B2], [BD]). [The citations 'B2' and 'BD' refer to Bôcher (1900–1901) and Bostan and Dumas (2010), respectively.]

References