PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] uniform convergence of integral (Definition)

Let the function $ f(x,\,t)$ be continuous in the domain

$\displaystyle a \leqq x < b, \,\, c \leqq t \leqq d,$
where $ b$ is a real number or $ \infty$, and let the improper integral
$\displaystyle F(t) = \int_a^b f(x,\,t)\,dx = \lim_{u \to b-}\int_a^u f(x,\,t)\,dx$ (1)

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
$\displaystyle \left\vert\int_x^b f(x,\,t)\,dx\right\vert < \varepsilon\quad\forall t\in [c,\,d]$
when $ x_\varepsilon \leqq x < b$.



"uniform convergence of integral" is owned by pahio.
(view preamble)

View style:

See Also: sum function of series

Also defines:  integral converging uniformly, uniformly converging integral

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: number, positive, interval, point, improper integral, real number, domain, continuous, function

This is version 9 of uniform convergence of integral, born on 2004-10-02, modified 2005-07-27.
Object id is 6277, canonical name is UniformConvergenceOfIntegral.
Accessed 4726 times total.

Classification:
AMS MSC26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)