In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class.
- A point is a degenerate circle, namely one with radius 0.
- A circle is a degenerate form of an ellipse, namely one with eccentricity 0.
- The line is a degenerate form of a parabola if the parabola resides on a tangent plane.
- A segment is a degenerate form of a rectangle, if this has a side of length 0.
- A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.
- A set containing a single point is a degenerate continuum.
- A random variable which can only take one value has a degenerate distribution.
- A sphere is a degenerate standard torus where the axis of revolution passes through the center of the generating circle, rather than outside it.
- A degenerate triangle has collinear vertices.
- See "general position" for other examples.
Similarly, roots of a polynomial are said to be degenerate if they coincide, since generically the n roots of an nth degree polynomial are all distinct. This usage carries over to eigenproblems: a degenerate eigenvalue (i.e. a multiply coinciding root of the characteristic polynomial) is one that has more than one linearly independent eigenvector.
In quantum mechanics any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels. Usually any such degeneracy indicates some underlying symmetry in the system.
For any non-empty subset , there is a bounded, axis-aligned degenerate rectangle
where and are constant (with for all ). The number of degenerate sides of is the number of elements of the subset . Thus, there may be as few as one degenerate "side" or as many as (in which case reduces to a singleton point).
- Degeneracy (graph theory)
- Vacuous truth
- Trivial (mathematics)
- Pathological (mathematics)
- Degenerate form