Improper Integral
is the limit of as ; at an isolated singularity , it is the limit of as . Otherwise it diverges.
Suppose is integrable on for every , and set
Then we say that exists (converges) if
then we set . Otherwise, we say that does not exist.
If is such that and , then we say exists, and set
Remark. This is different from
See further discussion in Example Sheet 4.
Isolated singularity
Let be integrable on for any . Set
Then, we say exists (converges) if exists and is finite, and we say
Otherwise, we say it does not exist (converge).
If is such that and , then say
Two-sided limits
The two-sided improper integral is defined by splitting at an interior point, not by a symmetric limit:
which is in general different from .
The same caution applies at an interior singularity:
and this is not equal to , as seen by taking .
Related
Stated in
- Definition 4.29 (Improper Integrals: Unbounded Domain)§4.5 Improper Integrals
- Definition 4.35 (Improper Integrals: Isolated Singularity)§4.5 Improper Integrals