# Parameter-dependent integral

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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 ).

## 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.

How to Cite This Entry:
Parameter-dependent integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parameter-dependent_integral&oldid=18531
This article was adapted from an original article by V.A. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article