Fuchs's theorem guarantees at least one Frobenius solution near a regular singular point
of a homogeneous linear second-order
ordinary differential equation. Its leading exponent
is a root of the indicial
equation, and need not be an integer.
If the two roots of the indicial equation do not differ by an integer, there are two
linearly independent Frobenius
solutions. If their difference is an integer, including
zero, a second linearly independent solution
may involve a logarithm. At an ordinary
point, two linearly independent solutions
can instead be expressed as Taylor series.
See also
Frobenius Method,
Frobenius Solution,
Indicial Equation,
Regular
Singular Point
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References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 462-463,
1985.National Institute of Standards and Technology. "Regular Singularities:
Fuchs-Frobenius Theory." §2.7(i) in Digital Library of Mathematical
Functions. https://dlmf.nist.gov/2.7#i.Referenced
on Wolfram|Alpha
Fuchs's Theorem
Cite this as:
Weisstein, Eric W. "Fuchs's Theorem."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FuchssTheorem.html
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