Improper Riemann Integrals
Riemann integral is defined only for bounded functions defined on a set of compact intervals.
Type 1
A function that is not integrable at one endpoint of the interval.
Suppose
Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.
Examples
The above integral converges (to
Type 2
A function defined on unbounded interval (including
Suppose
Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.
Examples
The above integral converges (to
Type 3
A function that is undefined at finite number of points. The integral can be
split into multiple integrals of type 1. Similarly integrals from
Convergence of improper integrals is similar to the convergence of series.