1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bounded functions with unbounded integrals

Tags:
  1. Apr 10, 2017 #1
    1. The problem statement, all variables and given/known data

    I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded).

    Note: The integrator is a system which gives an output equal to the anti-derivative of its input.

    3. The attempt at a solution

    I have already proven the instability of the differentiator by considering the bounded input ##f(t)= \sin(t^2),## which gives the unbounded output ##f'(t) = 2t \cos(t^2).##

    23h8yzp.jpg

    For the integrator, I know, for instance, that the bounded input ##f(t)=1## gives the unbounded output ##t.## But could anyone suggest a more interesting example like the one I gave for the differentiator?

    I couldn't come up with a good example. I would appreciate any suggestions or links.
     
  2. jcsd
  3. Apr 10, 2017 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your problem might be more interesting on a bounded interval. Think about ##\sqrt[3] x## on ##(0,1)## for a differentiation example. But on a bounded interval no example exists for the integration because ##\left | \int_a^t f(u)~du\right | \le \int_a^b |f(u)|~du\le M(b-a)## where ##M## is a bound for ##|f|## on ##[a,b]##.
     
  4. Apr 17, 2017 #3
    What about a non-bounded interval? I mean, I already found a function, ##f(t)=constant##, which increases indefinitely. The only other example I can think of, would be the Heaviside step function, ##f(t)=u(t)##. The output of the integrator for this function is:

    $$g(t) = \intop^t_{-\infty} u(\tau) d\tau = \intop^t_{-\infty} 1 \ d\tau = t$$

    for ##t>0.##

    So, is there really no other function whose integral increases indefinitely? :confused:
     
  5. Apr 17, 2017 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Of course not. There are lots of examples. Take any function that is nonnegative for ##x\ge 0## that is bounded but has an infinite area. One such example is ##\arctan x,~0\le x## which is bounded by ##\frac \pi 2##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted