is defined when is bounded and is a compact interval. If is any interval of and is bounded on every compact subset of (for example is continuous), then we can define the concept of improper integral of on by approximation of with compact sets.
Let with and and let be a continuous function on . Given we know that the function is Riemann integrable on ; hence we can define the improper integral of on as
Notice that the limit is taken in two variables. In the case when is compact this is the usual Riemann integral on (because the integral function is continuous). So there is no ambiguity in using the same simbol for improper integrals and usual Riemann integrals (but we will see that there is an ambiguity when dealing with Lebesgue integrals). Similarly, in the case when the interval is semi-open, i.e. when with , the definition clearly reduces to
which is a limit in one variable.
Notice also that an improper integral can be infinite or can possibly not exist (even thought is continuous).
The definition may be extended to the case when is the union of a finite number of intervals by summing up the improper integrals on every interval:
This limit can exist in some cases when the improper integral (not symmetric) fails to exist.
Given , one has:
Given , one can check that the improper integral exists and is finite but the improper integral is infinite. In particular this function is not summable (in the sense of Lebesgue integrals) on the interval .
The function has no improper integral in . But since one can easily check that the symmetric integral is zero:
|Date of creation||2013-03-22 12:30:06|
|Last modified on||2013-03-22 12:30:06|
|Last modified by||paolini (1187)|