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| In [[mathematics]], the '''Wronskian''' is a determinant introduced by {{harvs|txt|authorlink=Józef Maria Hoene-Wroński|first=Józef|last=Hoene-Wronski|year=1812}} and named by {{harvs|txt|authorlink=Thomas Muir (mathematician)|first=Thomas|last=Muir|year=1882|loc=Chapter XVIII}}. It is used in the study of [[differential equations]], where it can sometimes be used to show that a set of solutions is [[linear independence|linearly independent]].
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| ==Definition==
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| The Wronskian of two differentiable functions {{math|''f'' }} and {{math|''g''}} is {{math|''W''(''f'', ''g'') {{=}} ''f g''′ – ''g f'' ′}}.
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| More generally, for {{math|''n''}} [[real number|real]]- or [[complex number|complex]]-valued functions {{math|''f''<sub>1</sub>, . . . , ''f<sub>n</sub>''}}, which are {{math|''n'' – 1}} times [[differentiable]] on an [[interval (mathematics)|interval]] {{math|''I''}}, the Wronskian {{math|''W''(''f''<sub>1</sub>, . . . , ''f<sub>n</sub>'')}} as a function on {{math|''I''}} is defined by
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| :<math>
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| W(f_1, \ldots, f_n) (x)=
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| \begin{vmatrix}
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| f_1(x) & f_2(x) & \cdots & f_n(x) \\
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| f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\
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| \vdots & \vdots & \ddots & \vdots \\
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| f_1^{(n-1)}(x)& f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
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| \end{vmatrix},\qquad x\in I.
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| </math>
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| That is, it is the [[determinant]] of the [[matrix (math)|matrix]] constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the {{math|(''n'' – 1)}}th derivative, thus forming a [[square matrix]] sometimes called a '''fundamental matrix'''.
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| When the functions {{math|''f<sub>i</sub>''}} are solutions of a [[linear differential equation]], the Wronskian can be found explicitly using [[Abel's identity]], even if the functions {{math|''f<sub>i</sub>''}} are not known explicitly.
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| ==The Wronskian and linear independence==
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| If the functions {{math|''f<sub>i</sub>''}} are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is [[linear independence|linearly independent]] on an interval by showing that it does not vanish identically.
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| A common misconception is that {{math|''W'' {{=}} 0}} everywhere implies linear dependence, but {{harvtxt|Peano|1889}} pointed out that the functions {{math|''x''<sup>2</sup>}} and {{math|{{!}}''x''{{!}}''x''}} have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of {{math|0}}. There are several extra conditions which ensure that the vanishing of the Wronskian in an interval implies linear dependence.
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| {{harvtxt|Peano|1889}} observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent. {{harvtxt|Bocher|1901}} gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of {{math|''n''}} functions is identically zero and the {{math|''n''}} Wronskians of {{math|''n'' – 1}} of them do not all vanish at any point then the functions are linearly dependent. {{harvtxt|Wolsson|1989a}} gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.
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| ==Generalized Wronskians==
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| For {{math|''n''}} functions of several variables, a '''generalized Wronskian''' is the determinant of an {{math|''n''}} by {{math|''n''}} matrix with entries {{math|''D<sub>i</sub>''(''f<sub>j</sub>'')}} (with {{math|0 ≤ ''i'' < ''n''}}), where each {{math|''D<sub>i</sub>''}} is some constant coefficient linear partial differential operator of order {{math|''i''}}. If the functions are linearly dependent then all generalized Wronskians vanish. As in the 1 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 [[Thue–Siegel–Roth theorem|Roth's theorem]]. For more general conditions under which the converse is valid see {{harvtxt|Wolsson|1989b}}.
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| ==See also==
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| *[[Moore matrix]], analogous to the Wronskian with differentiation replaced by the [[Frobenius endomorphism]] over a finite field.
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| *[[Variation of parameters]]
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| ==References==
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| *{{Citation | last1=Bôcher | first1=Maxime | title=Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence | jstor=1986214 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | year=1901 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=2 | issue=2 | pages=139–149 | jfm=32.0313.02}}
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| *{{Citation | last1=Hartman | first1=Philip | title=Ordinary Differential Equations | url=http://books.google.com/books?id=CENAPMUEpfoC | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-89871-510-1 | mr=0171038 | zbl=0125.32102 | year=1964}}
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| *{{citation|first=J. |last=Hoene-Wronski|title=Réfutation de la théorie des fonctions analytiques de Lagrange|place= Paris |year=1812}}
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| *{{Citation | last1=Muir | first1=Thomas | title=A Treatise on the Theorie of Determinants. | url=http://www.archive.org/details/atreatiseontheo00muirgoog | publisher= Macmillan | year=1882 | jfm=15.0118.05}}
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| *{{Citation | last1=Peano | first1=Giuseppe | author1-link=Giuseppe Peano | title=Sur le déterminant wronskien. | language=French | jfm=21.0153.01 | year=1889 | journal=[[Mathesis (journal)|Mathesis]] | volume=IX | pages=75–76, 110–112}}
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| *{{eom|id=Wronskian|first=N. Kh. |last=Rozov}}
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| *{{Citation | last1=Wolsson | first1=Kenneth | title=A condition equivalent to linear dependence for functions with vanishing Wronskian | doi=10.1016/0024-3795(89)90393-5 | mr=989712 | zbl=0671.15005 | year=1989a | journal=Linear Algebra and its Applications | issn=0024-3795 | volume=116 | pages=1–8}}
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| *{{Citation | last1=Wolsson | first1=Kenneth | title=Linear dependence of a function set of {{math|''m''}} variables with vanishing generalized Wronskians | doi=10.1016/0024-3795(89)90548-X | mr=993032 | zbl=0724.15004 | year=1989b | journal=Linear Algebra and its Applications | issn=0024-3795 | volume=117 | pages=73–80}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Determinants]]
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| [[Category:Science and technology in Poland]]
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