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Thread: Logarithm integral

  1. #1
    Jem
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    Logarithm integral

    Evaluate $\displaystyle \int_0^1\frac{\ln(x+1)}x\,dx$.

    How many methods you know to solve it?
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  2. #2
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    Quote Originally Posted by Jem View Post
    Evaluate $\displaystyle \int_0^1\frac{\ln(x+1)}x\,dx$.

    How many methods you know to solve it?
    $\displaystyle \ln (x+1) = \sum_{n=1}^{\infty} \frac{(-1)^nx^n}{n}$.

    Thus, $\displaystyle \frac{\ln (x+1)}{x} = \sum_{n=1}^{\infty} \frac{(-1)^n x^{n-1}}{n}$.

    This means, $\displaystyle \int_0^1 \left( \sum_{n=1}^{\infty} \frac{(-1)^n x^{n-1}}{n}\right) dx = \sum_{n=1}^{\infty} \left( \int_0^1 \frac{(-1)^{n+1} x^{n-1}}{n} dx \right) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}= \frac{\pi^2}{12}$
    Last edited by ThePerfectHacker; Jan 1st 2008 at 08:43 PM.
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  3. #3
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    This particular integral is related to what is known as the 'dilogarithm'.

    $\displaystyle dilog(x)=\int_{1}^{x}\frac{ln(t)}{t-1}dt$

    In our case, $\displaystyle \int_{0}^{1}\frac{ln(x+1)}{x}dx=\int_{1}^{2}\frac{ ln(x)}{x-1}dx = -dilog(x+1) = \frac{{\pi}^{2}}{12}$

    It seems to me there is also a unit disk reference with the dilog. It says for n=2 and z in the unit disk, then

    $\displaystyle Li_{2}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{2}}$

    i.e, if z=1, then we have $\displaystyle \frac{{\pi}^{2}}{6}$

    There is also a trilogarithm.
    Last edited by galactus; Jan 1st 2008 at 02:33 PM.
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    what is dilog galactus?
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  5. #5
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    Do a google to find out more. It is a cool function with lotsa implications.

    For instance, it has the series: $\displaystyle -x-\frac{x^{2}}{4}-\frac{x^{3}}{9}-.....-\frac{x^{n}}{n^{2}}$

    Also, $\displaystyle Li_{2}(z)=\int_{z}^{0}\frac{ln(1-t)}{t}dt$

    In the event z = -1, then we have what PH posted.

    Notice how the Basel sum is involved here. $\displaystyle \sum_{k=1}^{\infty}\frac{1}{n^{2}}=\frac{{\pi}^{2} }{6}$

    I noticed the similarity between this and the posters integral and thought there was a connection. I am sure Wiki or Mathworld has something about it.

    The dilog is the polylog with n=2. There are others. As I mentioned, a trilog.

    It has implications to gamma and other things. Research it. You'll find it cool.
    Last edited by galactus; Jan 1st 2008 at 02:35 PM.
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  6. #6
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    Quote Originally Posted by ThePerfectHacker View Post
    $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}= \frac{\pi^2}{12}$
    $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
    {{n^2 }}} = 1 - \frac{1}
    {{2^2 }} + \frac{1}
    {{3^2 }} - \frac{1}
    {{4^2 }} + \cdots .$

    This last is equal to

    $\displaystyle \left( {1 + \frac{1}
    {{2^2 }} + \frac{1}
    {{3^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right) - 2\left( {\frac{1}
    {{2^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right).$

    Since $\displaystyle 2\left( {\frac{1}
    {{2^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right) = \frac{2}
    {{2^2 }}\left( {1 + \frac{1}
    {{2^2 }} + \cdots } \right),$

    we have $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
    {{n^2 }}} = \sum\limits_{n = 1}^\infty {\frac{1}
    {{n^2 }}} - \frac{1}
    {2}\sum\limits_{n = 1}^\infty {\frac{1}
    {{n^2 }}} = \frac{{\pi ^2 }}
    {6} - \frac{{\pi ^2 }}
    {{12}} = \frac{{\pi ^2 }}
    {{12}}.$

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  7. #7
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    Quote Originally Posted by Krizalid View Post
    $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
    {{n^2 }}} = 1 - \frac{1}
    {{2^2 }} + \frac{1}
    {{3^2 }} - \frac{1}
    {{4^2 }} + \cdots .$

    This last is equal to

    $\displaystyle \left( {1 + \frac{1}
    {{2^2 }} + \frac{1}
    {{3^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right) - 2\left( {\frac{1}
    {{2^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right).$

    Since $\displaystyle 2\left( {\frac{1}
    {{2^2 }} + \frac{1}
    {{4^2 }} + \cdots } \right) = \frac{2}
    {{2^2 }}\left( {1 + \frac{1}
    {{2^2 }} + \cdots } \right),$

    we have $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
    {{n^2 }}} = \sum\limits_{n = 1}^\infty {\frac{1}
    {{n^2 }}} - \frac{1}
    {2}\sum\limits_{n = 1}^\infty {\frac{1}
    {{n^2 }}} = \frac{{\pi ^2 }}
    {6} - \frac{{\pi ^2 }}
    {{12}} = \frac{{\pi ^2 }}
    {{12}}.$

    That is the standard way this identity is derived from $\displaystyle \zeta (2)$.
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