Parameter-dependent integral
An integral of the type
![]() |
in which the point ranges over the space
(if the point ranges only over a certain domain
in
, the function
may be assumed to vanish for
), while the point
, representing a set of parameters
, varies within some domain
of the space
.
The main concern of the theory of such integrals is to determine conditions for the continuity and differentiability of with respect to the parameters
. If
is interpreted as a Lebesgue integral, one obtains less restrictive conditions for its continuity and differentiability. The following propositions are valid.
1) If is continuous in
in the domain
for almost-all
and if there exists an integrable function
on
such that
for every
and almost-all
, then
is continuous in
.
2) Let be a function defined for
,
. Assume that the derivative
exists for almost-all
and every
and that is a continuous function of
on
for almost-all
. Assume, moreover, that there exists an integrable function
on
such that
for every
and almost-all
. Finally, assume that for some
the integral
![]() |
exists. Then the function
![]() |
is differentiable with respect to on
, and its derivative
may be evaluated by differentiating under the integral sign:
![]() |
These two propositions imply a series of simpler propositions about the continuity and differentiability of integrals with parameters, relating to the interpretation of the integral as a Riemann integral and to more specific cases (see [2]–[4]).
Parameter-dependent improper integrals.
For the simplest improper integral of the first kind,
![]() | (*) |
one introduces the notion of uniform convergence with respect to the parameter in a closed interval
. This integral is said to be uniformly convergent in
on
if, for each
, there exists an
such that
![]() |
for all .
The following propositions are valid.
a) If is continuous in a half-strip
and if the integral (*) is uniformly convergent in
on
, then
is continuous in
.
b) If and the derivative
are continuous in a half-strip
, if the integral (*) is convergent for some
and if the integral
![]() |
is uniformly convergent in on
, then the function
is differentiable on
and its derivative may be evaluated by the formula
![]() |
Analogous propositions hold for improper integrals of the second kind.
References
[1] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) |
[2] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 2 , MIR (1982) (Translated from Russian) |
[3] | L.D. Kudryavtsev, "Mathematical analysis" , 2 , Moscow (1970) (In Russian) |
[4] | S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian) |
[5] | A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) |
Comments
The propositions stated are simple consequences of Lebesgue's dominated convergence principle (see Lebesgue theorem 2)).
Parameter-dependent integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parameter-dependent_integral&oldid=32839