In mathematics, the Wronskian of <math>n</math> differentiable functions is the determinant of a matrix formed by the functions and their derivatives up to order <math>n-1</math>. It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can show the linear independence of certain sets of solutions.
Definition
The Wrońskian of two differentiable functions <math>f</math> and <math>g</math> is <math>W(f,g)=f g' - g f'</math>.
More generally, for <math>n</math> real- or complex-valued functions <math>f_1,\dots,f_n</math>, which are <math>n-1</math> times differentiable on an interval <math>I</math>, the Wronskian <math>W(f_1,\ldots,f_n)</math> is the function
:<math> W(f_1, \ldots, f_n) (x)=\begin{vmatrix}
f_1(x) & f_2(x) & \cdots & f_n(x) \\
f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\
\vdots & \vdots & \ddots & \vdots \\
f_1^{(n-1)}(x)& f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}
</math>
defined for all <math>x\in I</math>.
This is the determinant of the matrix constructed by placing the functions in the first row, their first derivatives of the functions in the second row, and so on through the <math>(n-1)</math>-st derivative, thus forming a square matrix.
When the functions are solutions of a linear differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions themselves are not known explicitly. (See below.)
The Wronskian and linear independence
If the functions 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.
A common misconception is that <math>W=0</math> everywhere implies linear dependence. pointed out that the functions and have continuous derivatives and their Wrońskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of .