Improper Integral
المؤلف:
Jeffreys, H. and Jeffreys, B. S.
المصدر:
"Infinite and Improper Integrals." §1.104 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
21-8-2018
2703
Improper Integral
An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.
For example, the integral
 |
(1)
|
is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values
 |
(2)
|
and then taking the limit as 
,
Improper integrals of the form
 |
(6)
|
with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If 
decreases at least as fast as 
, then let
and
If 
diverges as 
for ![gamma in [0,1]](http://mathworld.wolfram.com/images/equations/ImproperIntegral/Inline33.gif)
, let
and
 |
(18)
|
If 
diverges as 
for ![gamma in [0,1]](http://mathworld.wolfram.com/images/equations/ImproperIntegral/Inline51.gif)
, let
and
If the integral diverges exponentially, then let
and
 |
(27)
|
REFERENCES:
Jeffreys, H. and Jeffreys, B. S. "Infinite and Improper Integrals." §1.104 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 33-34, 1988.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Improper Integrals." §4.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 135-140, 1992.
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