Results 1 to 2 of 2

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

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

    Further, prove that,
    $\displaystyle \int^{\infty}_af(x)dx+\int^a_{-\infty}f(x)dx=\int^{\infty}_bf(x)dx+\int^b_{-\infty}f(x)dx$.
    Thus, proving that the value of
    $\displaystyle \int^{\infty}_{-\infty}f(x)dx$
    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,
    $\displaystyle \int^{\infty}_{b}+\int^{b}_{-\infty}$ if $\displaystyle a<b$ then write $\displaystyle b=a+r,r>0$. Thus, we have,
    $\displaystyle \int^{\infty}_{a+r}+\int^{a+r}_{-\infty}$
    By the subdivision rule,
    $\displaystyle \int_{-\infty}^a+\int^{a+r}_{a}+\int^a_{a+r}+\int_a^{ \infty }$
    But, $\displaystyle \int^{a+r}_{a}+\int^a_{a+r}=0$
    (Same when $\displaystyle a>b$)
    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: Aug 7th 2010, 11:27 AM
  2. improper integral help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 19th 2010, 02:24 PM
  3. improper integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 25th 2009, 07:06 PM
  4. Improper Integral
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Mar 4th 2009, 01:52 PM
  5. improper integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 9th 2008, 08:32 PM

Search Tags


/mathhelpforum @mathhelpforum