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Difference between revisions of "Parameter-dependent integral"

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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|Lebesgue integral]], one obtains less restrictive conditions for its continuity and differentiability. The following propositions are valid.
 
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|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 $G$.
+
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
 
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

Revision as of 14:50, 11 August 2014

An integral of the type

$$J(y)=\int f(x,y)dx,$$

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 '"`UNIQ-MathJax2-QINU`"' exists. Then the function '"`UNIQ-MathJax3-QINU`"' is differentiable with respect to $t$ on $(a,b)$, and its derivative $J'(t)$ may be evaluated by differentiating under the integral sign: '"`UNIQ-MathJax4-QINU`"' 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 [[#References|[2]]]–[[#References|[4]]]). =='"`UNIQ--h-0--QINU`"'Parameter-dependent improper integrals.== For the simplest [[Improper integral|improper integral]] of the first kind, '"`UNIQ-MathJax5-QINU`"' 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 '"`UNIQ-MathJax6-QINU`"' 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 \ref{*} 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 \ref{*} is convergent for some $t\in[c,d]$ and if the integral '"`UNIQ-MathJax7-QINU`"' 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

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

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=32839
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