Wronskian of two functions

Definition

Suppose $f$ and $g$ are both functions of one variable. The Wronskian of $f$ and $g$ is defined as the determinant of the following matrix:

$\left(\begin{array}{cc}f& g\\ f\text{'}& g\text{'}\\ \end{array}\right)$

Explicitly, it is the function:

$x\mapsto f(x)g^{\prime }(x)-f^{\prime }(x)g(x)$

defined wherever the right side expression makes sense, which happens at the points where $f,g$ and their derivatives $f^{\prime },g^{\prime }$ exist.

Note that the Wronskian is skew-symmetric in $f$ and $g$ rather than symmetric, i.e., the Wronskian of $g$ and $f$ is the negative of the Wronskian of $f$ and $g$. We are typically concerned, not with the precise Wronskian but with the Wronskian up to scalar multiples, and in particular with whether it is identically zero. These aspects of its behavior are symmetric.