uniform convergence of integralUniformConvergenceOfIntegral2004-10-02 15:04:382009-02-11 10:59:39Definitionimproper integralintegral converging uniformlyuniformly converging integral% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet the function\, $f(x,\,t)$\, be continuous in the domain
$$a \;\leqq\; x \;<\; b, \quad c \;\leqq\; t \;\leqq\; d,$$
where $b$ is a real number or $\infty$, and let the improper integral
\begin{align}
F(t) \;:=\, \int_a^b f(x,\,t)\,dx \;=\; \lim_{u \to b-}\int_a^u f(x,\,t)\,dx
\end{align}
be \PMlinkname{convergent}{ImproperIntegral} in every point $t$ of the interval\, $[c,\,d]$.\, We say that the \PMlinkescapetext{{\em integral converges uniformly}} on the interval\, $[c,\,d]$,\, if for each positive number $\varepsilon$ there is a value $x_\varepsilon \in [a,\,b]$\, such that
$$\left|\int_x^b f(x,\,t)\,dx\right| \;<\; \varepsilon\quad\forall t\in [c,\,d]$$
when\, $x_\varepsilon \leqq x < b$.