Results 1 to 2 of 2

Math Help - Uniqueness of Improper Integral

  1. #1
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10

    Uniqueness of Improper Integral

    If f(x) is a countinous real function, we define,
    \int^{\infty}_{-\infty}f(x)dx as being convergent if there exists a a\in\mathbb{R} such as, \int^{\infty}_af(x)dx and \int^a_{-\infty}f(x)dx are convergent. Also its value is then,
    \int^{\infty}_{-\infty}f(x)dx=\int^{\infty}_af(x)dx+\int^a_{-\infty}f(x)dx .

    Prove that if,
    \int^{\infty}_{-\infty}f(x)dx is convergent then for any b\in\mathbb{R} the two improper integrals,
    \int^{\infty}_bf(x)dx \mbox{ and }\int^b_{-\infty}f(x)dx are also convergent.

    Further, prove that,
    \int^{\infty}_af(x)dx+\int^a_{-\infty}f(x)dx=\int^{\infty}_bf(x)dx+\int^b_{-\infty}f(x)dx.
    Thus, proving that the value of
    \int^{\infty}_{-\infty}f(x)dx
    is well-defined.
    ???
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Thank you but I think I have it. Basically, you rewrite your integral.
    The integral,
    \int^{\infty}_{b}+\int^{b}_{-\infty} if a<b then write b=a+r,r>0. Thus, we have,
    \int^{\infty}_{a+r}+\int^{a+r}_{-\infty}
    By the subdivision rule,
    \int_{-\infty}^a+\int^{a+r}_{a}+\int^a_{a+r}+\int_a^{ \infty }
    But, \int^{a+r}_{a}+\int^a_{a+r}=0
    (Same when a>b)
    And the proof it complete.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Improper integral
    Posted in the Calculus Forum
    Replies: 13
    Last Post: August 7th 2010, 12:27 PM
  2. improper integral help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 19th 2010, 03:24 PM
  3. improper integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 25th 2009, 08:06 PM
  4. Improper Integral
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 4th 2009, 02:52 PM
  5. improper integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 9th 2008, 09:32 PM

Search Tags


/mathhelpforum @mathhelpforum