Definition:Wronskian

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Definition

Let $f \left({x}\right)$ and $g \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Let $f$ and $g$ be differentiable on $\left[{a \,.\,.\, b}\right]$.


The Wronskian of $f$ and $g$ is defined as:

$W \left({f, g}\right) = \begin{vmatrix} f \left({x}\right) & g \left({x}\right) \\ f' \left({x}\right) & g' \left({x}\right) \\ \end{vmatrix} = f \left({x}\right) g' \left({x}\right) - g \left({x}\right) f' \left({x}\right)$


General Definition

Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Let $f_1, f_2, \ldots, f_n$ be $n-1$ times differentiable on $\left[{a \,.\,.\, b}\right]$.


The Wronskian of $f_1, f_2, \ldots, f_n$ on $\left[{a \,.\,.\, b}\right]$ is defined as:

$W \left({f_1, f_2, \ldots, f_n}\right) = \begin{vmatrix} f_1 \left({x}\right) & f_2 \left({x}\right) & \cdots & f_n \left({x}\right) \\ {f_1}' \left({x}\right) & {f_2}' \left({x}\right) & \cdots & {f_n}' \left({x}\right) \\ \vdots & \vdots & \ddots & \vdots \\ {f_1}^{\left({n - 1}\right)} \left({x}\right) & {f_2}^{\left({n - 1}\right)} \left({x}\right) & \cdots & {f_n}^{\left({n - 1}\right)} \left({x}\right) \\ \end{vmatrix}$

where:

$\begin{vmatrix} \cdots \end{vmatrix}$ denotes the determinant
${f_1}^{\left({n - 1}\right)} \left({x}\right)$ denotes the $n-1$th derivative of $f_1$.


Also known as

Some sources preserve the diacritic on the n, that is: Wrońskian, but many consider such refinements to be visual clutter and prefer to discard them.


Source of Name

This entry was named for Józef Maria Hoene-Wroński.


Sources