Improper integral

A definite integral whose interval is unbounded or whose integrand is unbounded or undefined at some point, defined through a limit.

Limit: A value that a function or sequence approaches as its input approaches a point or infinity. The limit defines an improper integral by moving a cutoff toward the problematic point.
Definite integral: A number representing the signed accumulation of a function over an interval, commonly defined by a limit of sums. Improper integrals extend this finite-interval construction to unbounded or singular cases.
Probability density function: A nonnegative function whose integral over a region gives the probability assigned to that region. Densities on unbounded supports must have a finite total integral to define probability distributions.
Lebesgue integral: An integral defined by measuring the values a function takes over measurable sets. It handles many functions beyond the Riemann framework without relying on endpoint cutoff limits.
Infinite interval: An interval that extends without bound in at least one direction. An unbounded integration interval is handled by sending a finite endpoint to infinity.
Riemann integral: An integral defined by limits of sums over partitions of an interval. The ordinary Riemann integral supplies the finite, bounded case that improper definitions extend.
Gamma function: A function extending the factorial to many real and complex arguments through an integral representation. Its defining integral converges on an unbounded interval despite an endpoint singularity for some parameters.
Divergent integral: An attempted integral whose defining limit does not exist as a finite number. Divergence is the outcome when the improper integral's required limit fails.
Cauchy principal value: A symmetric limiting value assigned to certain singular integrals, even when the ordinary improper integral diverges. Symmetric cancellation can produce a principal value without making the improper integral converge.
Convergence: The property of approaching a finite value or a specified limit. An improper integral exists only when its defining limit converges.

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