Parameter-dependent integral

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)).