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An integral of the type
 
An integral of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714701.png" /></td> </tr></table>
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$$J(y)=\int f(x,y)\,dx,$$
  
in which the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714702.png" /> ranges over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714703.png" /> (if the point ranges only over a certain domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714704.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714705.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714706.png" /> may be assumed to vanish for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714707.png" />), while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714708.png" />, representing a set of parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p0714709.png" />, varies within some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147010.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147011.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147012.png" /> with respect to the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147014.png" /> is interpreted as a [[Lebesgue integral|Lebesgue integral]], one obtains less restrictive conditions for its continuity and differentiability. The following propositions are valid.
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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.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147015.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147016.png" /> in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147017.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147018.png" /> and if there exists an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147021.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147022.png" /> and almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147024.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147025.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147026.png" /> be a function defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147028.png" />. Assume that the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147029.png" /> exists for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147030.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147031.png" /> and that is a continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147033.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147034.png" />. Assume, moreover, that there exists an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147037.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147038.png" /> and almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147039.png" />. Finally, assume that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147040.png" /> the integral
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147041.png" /></td> </tr></table>
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$$\int f(x,t_0)\,dx$$
  
 
exists. Then the function
 
exists. Then the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147042.png" /></td> </tr></table>
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$$J(t)=\int f(x,t)\,dx$$
  
is differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147044.png" />, and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147045.png" /> may be evaluated by differentiating under the integral sign:
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is differentiable with respect to $t$ on $(a,b)$, and its derivative $J'(t)$ may be evaluated by differentiating under the integral sign:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147046.png" /></td> </tr></table>
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$$J'(t)=\int\frac{\partial f}{\partial t}(x,t)\,dx.$$
  
 
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 [[#References|[2]]]–[[#References|[4]]]).
 
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]]]).
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For the simplest [[Improper integral|improper integral]] of the first kind,
 
For the simplest [[Improper integral|improper integral]] of the first kind,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$J(t)=\int\limits_a^\infty f(x,t)\,dx,\label{*}\tag{*}$$
  
one introduces the notion of uniform convergence with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147048.png" /> in a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147049.png" />. This integral is said to be uniformly convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147051.png" /> if, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147052.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147053.png" /> such that
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147054.png" /></td> </tr></table>
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$$\left|\int\limits_R^\infty f(x,t)\,dx\right|<\epsilon$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147055.png" />.
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for all $R\geq A(\epsilon)$.
  
 
The following propositions are valid.
 
The following propositions are valid.
  
a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147056.png" /> is continuous in a half-strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147057.png" /> and if the integral (*) is uniformly convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147060.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147061.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147062.png" /> and the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147063.png" /> are continuous in a half-strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147064.png" />, if the integral (*) is convergent for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147065.png" /> and if the integral
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147066.png" /></td> </tr></table>
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$$\int\limits_a^\infty\frac{\partial f}{\partial t}(x,t)\,dx$$
  
is uniformly convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147068.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147069.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147070.png" /> and its derivative may be evaluated by the formula
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147071.png" /></td> </tr></table>
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$$J'(t)=\int\limits_a^\infty\frac{\partial f}{\partial t}(x,t)\,dx.$$
  
 
Analogous propositions hold for improper integrals of the second kind.
 
Analogous propositions hold for improper integrals of the second kind.

Latest revision as of 17:33, 14 February 2020

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

$$\int f(x,t_0)\,dx$$

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=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