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)).