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Parameter-dependent integral

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An integral of the type

in which the point x=(x_1,\ldots,x_n) ranges over the space \mathbf R^n (if the point ranges only over a certain domain D in \mathbf R^n, the function f(x,y) may be assumed to vanish for x\in\mathbf R^n\setminus D), while the point y=(y_1,\ldots,y_m), representing a set of parameters y_1,\ldots,y_m, varies within some domain G of the space \mathbf R^m.

The main concern of the theory of such integrals is to determine conditions for the continuity and differentiability of J(y) with respect to the parameters y_1,\ldots,y_m. If J(y) is interpreted as a Lebesgue integral, one obtains less restrictive conditions for its continuity and differentiability. The following propositions are valid.

1) If f(x,y) is continuous in y in the domain G\subset\mathbf R^m for almost-all x\in\mathbf R^n and if there exists an integrable function g on \mathbf R^n such that |f(x,y)|\leq g(x) for every y\in G and almost-all x\in\mathbf R^n, then J(y) is continuous in G.

2) Let f(x,t) be a function defined for x\in\mathbf R^n, t\in(a,b). Assume that the derivative \partial f(x,t)/\partial t exists for almost-all x\in\mathbf R^n and every t\in(a,b) and that is a continuous function of t on (a,b) for almost-all x\in\mathbf R^n. Assume, moreover, that there exists an integrable function g on \mathbf R^n such that |\partial f(x,t)/\partial t|\leq g(x) for every t\in(a,b) and almost-all x\in\mathbf R^n. Finally, assume that for some t_0\in(a,b) the integral

\int f(x,t_0)\,dx

exists. Then the function

J(t)=\int f(x,t)\,dx

is differentiable with respect to t on (a,b), and its derivative J'(t) may be evaluated by differentiating under the integral sign:

J'(t)=\int\frac{\partial f}{\partial t}(x,t)\,dx.

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,

J(t)=\int\limits_a^\infty f(x,t)\,dx,\label{*}\tag{*}

one introduces the notion of uniform convergence with respect to the parameter t in a closed interval c\leq t\leq d. This integral is said to be uniformly convergent in t on [c,d] if, for each \epsilon>0, there exists an A(\epsilon)>0 such that

\left|\int\limits_R^\infty f(x,t)\,dx\right|<\epsilon

for all R\geq A(\epsilon).

The following propositions are valid.

a) If f(x,t) is continuous in a half-strip [a\leq x<\infty,c<t\leq d] and if the integral \eqref{*} is uniformly convergent in t on [c,d], then J(t) is continuous in c<t\leq d.

b) If f(x,t) and the derivative \partial f(x,t)/\partial t are continuous in a half-strip [a\leq x<\infty,c\leq t\leq d], if the integral \eqref{*} is convergent for some t\in[c,d] and if the integral

\int\limits_a^\infty\frac{\partial f}{\partial t}(x,t)\,dx

is uniformly convergent in t on [c,d], then the function J(t) is differentiable on [c,d] and its derivative may be evaluated by the formula

J'(t)=\int\limits_a^\infty\frac{\partial f}{\partial t}(x,t)\,dx.

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)).

How to Cite This Entry:
Parameter-dependent integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parameter-dependent_integral&oldid=44764
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