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uniform convergence of integral
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(Definition)
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Let 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
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be convergent in every point $t$ of the interval $[c,\,d]$ . We say that the 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 $x_\varepsilon \leqq x
< b$ .
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"uniform convergence of integral" is owned by pahio.
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Cross-references: number, positive, interval, point, improper integral, real number, domain, continuous, function
There is 1 reference to this entry.
This is version 10 of uniform convergence of integral, born on 2004-10-02, modified 2009-02-11.
Object id is 6277, canonical name is UniformConvergenceOfIntegral.
Accessed 5909 times total.
Classification:
| AMS MSC: | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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