A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the point is a degenerate case of the circle as the radius approaches 0, and the circle is a degenerate form of an ellipse as the eccentricity approaches 0. Another example is the two identical roots of the second-order polynomial (x-1)^2. Since the n roots of an nth degree polynomial are usually distinct, roots which coincide are said to be degenerate. Degenerate cases often require special treatment in numerical and analytical solutions. For example, a simple search for both roots of the above equation would find only a single one: 1.

The word degenerate also has several very specific and technical meanings in different branches of mathematics.

See also

Monster-Barring, Multiple Root, Pathological, Trivial

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Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 513-514, 1985.

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Cite this as:

Weisstein, Eric W. "Degenerate." From MathWorld--A Wolfram Web Resource.

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