Difference between revisions of "Divergent integral"
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| − | + | A concept opposite to that of a convergent integral (see also [[Singular integral|Singular integral]]). For example, if a function $ f $ | |
| + | is defined on a bounded or unbounded interval $ [ a, b) $, | ||
| + | $ - \infty \leq a \leq b \leq \infty $, | ||
| + | if for each $ \eta \in [ a, b) $ | ||
| + | it is integrable on $ [ a, \eta ] $ | ||
| + | and if there is no finite limit | ||
| − | + | $$ | |
| + | \lim\limits _ {\eta \rightarrow b } \ | ||
| + | \int\limits _ { a } ^ \eta | ||
| + | f ( x) dx, | ||
| + | $$ | ||
| − | one says that the divergent integral | + | then one says that the integral $ \int _ {a} ^ {b} f ( x) dx $ |
| + | diverges. In the case that | ||
| + | |||
| + | $$ | ||
| + | \lim\limits _ {\eta \rightarrow b } \ | ||
| + | \int\limits _ { a } ^ \eta | ||
| + | f ( x) dx = \ | ||
| + | + \infty \ \textrm{ or } \ \ | ||
| + | - \infty , | ||
| + | $$ | ||
| + | |||
| + | one says that the divergent integral $ \int _ {a} ^ {b} f ( x) dx $ | ||
| + | is equal to $ + \infty $ | ||
| + | or $ - \infty $, | ||
| + | respectively. | ||
Latest revision as of 19:36, 5 June 2020
A concept opposite to that of a convergent integral (see also Singular integral). For example, if a function $ f $
is defined on a bounded or unbounded interval $ [ a, b) $,
$ - \infty \leq a \leq b \leq \infty $,
if for each $ \eta \in [ a, b) $
it is integrable on $ [ a, \eta ] $
and if there is no finite limit
$$ \lim\limits _ {\eta \rightarrow b } \ \int\limits _ { a } ^ \eta f ( x) dx, $$
then one says that the integral $ \int _ {a} ^ {b} f ( x) dx $ diverges. In the case that
$$ \lim\limits _ {\eta \rightarrow b } \ \int\limits _ { a } ^ \eta f ( x) dx = \ + \infty \ \textrm{ or } \ \ - \infty , $$
one says that the divergent integral $ \int _ {a} ^ {b} f ( x) dx $ is equal to $ + \infty $ or $ - \infty $, respectively.
How to Cite This Entry:
Divergent integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergent_integral&oldid=17382
Divergent integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergent_integral&oldid=17382
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article