The Wronskian of 
functions 
, ..., 
that are 
times differentiable
is the determinant
If the Wronskian is nonzero at a point of an interval, the functions are linearly
independent on that interval. An identically vanishing
Wronskian implies linear dependence
on some subinterval, but need not imply linear
dependence on the whole interval. For analytic
functions on a connected interval, the Wronskian
vanishes identically iff the functions
are linearly dependent (Bostan and
Dumas 2010, p. 722).
The Wronskian is implemented in the Wolfram Language as Wronskian[list,
x].
See also
Abel's Differential Equation Identity,
Gram Determinant,
Hessian,
Jacobian,
Linearly
Dependent Functions
Explore with Wolfram|Alpha
References
Bostan, A. and Dumas, P. "Wronskians and Linear Independence." Amer. Math. Monthly 117, 722-727, 2010. https://doi.org/10.4169/000298910X515785.Gradshteyn,
I. S. and Ryzhik, I. M. "Wronskian Determinants." §14.315
in Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
p. 1069, 2000.Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 524-525,
1953.Referenced on Wolfram|Alpha
Wronskian
Cite this as:
Weisstein, Eric W. "Wronskian." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Wronskian.html
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