View source for Parameter-dependent integral

An integral of the type

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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" />.

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.

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" />.

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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exists. Then the function

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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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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]]]).

==Parameter-dependent improper integrals.==
For the simplest [[Improper integral|improper integral]] of the first kind,

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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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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071470/p07147055.png" />.

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" />.

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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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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Analogous propositions hold for improper integrals of the second kind.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Kudryavtsev,   "Mathematical analysis" , '''2''' , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>



====Comments====
The propositions stated are simple consequences of Lebesgue's dominated convergence principle (see [[Lebesgue theorem|Lebesgue theorem]] 2)).