Results 1 to 11 of 11

Math Help - Lebesgue integral of Odd Function.

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    45

    Lebesgue integral of Odd Function.

    Suppose i have f is an odd function.
    How to show that  \int_{[-\pi,\pi]} f(x) =0 ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by younhock View Post
    Suppose i have f is an odd function.
    How to show that  \int_{[-\pi,\pi]} f(x) =0 ?
    This is true if f(x) has no singularities in [-\pi,\pi]... consider for example f(x)=\frac{1}{x}...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2010
    Posts
    45

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by chisigma View Post
    This is true if f(x) has no singularities in [-\pi,\pi]... consider for example f(x)=\frac{1}{x}...

    Kind regards

    \chi \sigma
    Is there any way to show it? Thank you.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Lebesgue integral of Odd Function.

    An odd function that is not the null function is positive in a part and negative in a different part of the definition interval. Let's define f^{+} (x)= f(x) where f(x)>0 and f^{+} (x)= 0 elsewhere and f^{-} (x)= -f(x) where f(x)<0 and f^{-} (x)= 0 elsewhere. The Lebesgue integral of f(x) is defined as...

    \int_{\mu} f(x) dx = \int_{\mu} f^{+}(x) dx - \int_{\mu} f^{-}(x) dx (1)

    Now the integral (1) exists only if both the integrals in (1) exist and that is not true [for example...] for f(x)= \frac{1}{x} in [- \pi, +\pi]...



    Marry Christmas from Serbia

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Lebesgue integral of Odd Function.

    Well, \frac{1}{x} isn't Lebesgue integrable on [-\pi,\pi], so it's a moot point, no? The basic idea is that change of variables still works for Lebsgue integrals. What if you let x=-z?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by Drexel28 View Post
    Well, \frac{1}{x} isn't Lebesgue integrable on [-\pi,\pi], so it's a moot point, no? The basic idea is that change of variables still works for Lebsgue integrals. What if you let x=-z?
    I'm very sorry but I'm afraid not to have fully undestood Your reply... can You explain a little more, please?...



    Marry Christmas from Serbia

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by chisigma View Post
    I'm very sorry but I'm afraid not to have fully undestood Your reply... can You explain a little more, please?...



    Marry Christmas from Serbia

    \chi \sigma
    Which part of it? The fact that \frac{1}{x} isn't Lebesgue integrable, or how the change of variables works?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by Drexel28 View Post
    Which part of it? The fact that \frac{1}{x} isn't Lebesgue integrable, or how the change of variables works?
    The original question was to demonstrate that if f(x) is and odd function, then ...

    \int_{-\pi}^{+\pi} f(x) dx =0 (1)

    ... where the integral is 'Lebesgue Integral'. My replay has been [symply...] that an odd function isn't necessarly Lebesgue integrable and the example f(x)= \frac{1}{x} has been supplied. Of course if we add the condition that f(x) must be 'regular', then the demonstration of (1) is very comfortable...



    Marry Christmas from Serbia

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by chisigma View Post
    The original question was to demonstrate that if f(x) is and odd function, then ...

    \int_{-\pi}^{+\pi} f(x) dx =0 (1)

    ... where the integral is 'Lebesgue Integral'. My replay has been [symply...] that an odd function isn't necessarly Lebesgue integrable and the example f(x)= \frac{1}{x} has been supplied. Of course if we add the condition that f(x) must be 'regular', then the demonstration of (1) is very comfortable...



    Marry Christmas from Serbia

    \chi \sigma
    Clearly we are to assume that the function is integrable, otherwise the integral doesn't make sense! That's like saying "Prove that the limit of a sequence in [0,\infty) is in [0,\infty) is false since \lim n is infinite--it's clear by wording that we are to assume the sequence converges!
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Mar 2010
    Posts
    45

    Re: Lebesgue integral of Odd Function.

    Maybe I should add one more condition to the question since it causes confusion. Now f(x) is lebesgue integrable and odd. how to show the integral value is zero?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Lebesgue integral of Odd Function.

    Quote Originally Posted by younhock View Post
    Maybe I should add one more condition to the question since it causes confusion. Now f(x) is lebesgue integrable and odd. how to show the integral value is zero?
    ... if f(x) is odd then f(x)=-f(-x) so that is...

    \int_{-\pi}^{0} f(x)\ dx = - \int_{0}^{\pi} f(x)\ dx \implies  \int_{-\pi}^{0} f(x)\ dx + \int_{0}^{\pi} f(x)\ dx = 0



    Marry Christmas from Serbia

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 29th 2010, 03:00 AM
  2. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 22nd 2010, 09:36 AM
  3. Lebesgue integral
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: January 9th 2010, 06:01 PM
  4. Lebesgue Integral
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: December 22nd 2009, 06:35 PM
  5. Lebesgue integral
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 24th 2009, 01:24 AM

Search Tags


/mathhelpforum @mathhelpforum