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Math Help - Gauss integral

  1. #1
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    Gauss integral

    where $ is the integral symbol , how can i get a general solution to

    the indefinite integral $ dx / ln(x) .

    problem relates to the primes.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Demus View Post
    where $ is the integral symbol , how can i get a general solution to

    the indefinite integral $ dx / ln(x) .

    problem relates to the primes.
    \int_a^{b}\frac{1}{\ln(x)}dx
    Let ln(x)=u\Rightarrow{x=e^{u}}
    Then dx=e^{u}
    new limits of integration are ln(a) and ln(b)

    So we have \int_{\ln(a)}^{\ln(b)}\frac{e^{u}}{u}du

    this is the exponential integral and has the power series solution

    \sum_{n=0}^{\infty}\frac{x^n}{n\cdot{n!}}
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Mathstud28 View Post
    \int_a^{b}\frac{1}{\ln(x)}dx
    Let ln(x)=u\Rightarrow{x=e^{u}}
    Then dx=e^{u}
    new limits of integration are ln(a) and ln(b)

    So we have \int_{\ln(a)}^{\ln(b)}\frac{e^{u}}{u}du

    this is the exponential integral and has the power series solution

    \sum_{n=0}^{\infty}\frac{x^n}{n\cdot{n!}}
    Sloppy work, how can a definite integral have a power series expansion like this.

    Also, the OP asked for an indefinite integral. Which should be related to logarithmic integral as well as the exponential integral, but you dont get there this way.

    RonL
    Last edited by CaptainBlack; April 30th 2008 at 11:45 AM.
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  4. #4
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    integrals

    thanks M28 , your reply was helpful as it does form part of a solution
    as far as mathcad is concerned .there does seem to be more to it which i do
    not understand yet , involving Ei , the exponential integral .I wonder if captain black can explain this ?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Demus View Post
    thanks M28 , your reply was helpful as it does form part of a solution
    as far as mathcad is concerned .there does seem to be more to it which i do
    not understand yet , involving Ei , the exponential integral .I wonder if captain black can explain this ?
    Sloppy work on my part I of course meant the logarithmic integral:

    {\rm{li}}(x)=\int_0^x \frac{1}{\ln(t)}~dt

    or the offset logarithmic integral (to aviod the singularity at 1):

     <br />
{\rm{Li}}(x)=\int_2^x \frac{1}{\ln(t)}~dt<br />

    But also the exponential integral:

    \int_1^x \frac{1}{\ln(t)}~dt=\int_0^{\ln(x)}\frac{e^u}{u}~d  u={\rm{Ei}}(\ln(x))-{\rm{Ei}}(0)

    Now this has not been as careful as I would like so still probably needs checking some more.

    See the Wikipedia artice on the logarithmic integral for more information.

    RonL
    Last edited by CaptainBlack; April 30th 2008 at 11:56 AM.
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  6. #6
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    Li

    I would like to show symbols . As i am very newbie , i'm in need of tutorials so as to better illustrate my questions .

    I came across the integral when reading about Guass . It looked harmless untill i tried to integrate . Eventually i gave up and solved it symbolic using mathcad .Then i realised i was in over my head with the Ei.

    My next quest is to understand how Ei is derived .
    many thanks .
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Demus View Post
    I would like to show symbols . As i am very newbie , i'm in need of tutorials so as to better illustrate my questions .

    I came across the integral when reading about Guass . It looked harmless untill i tried to integrate . Eventually i gave up and solved it symbolic using mathcad .Then i realised i was in over my head with the Ei.

    My next quest is to understand how Ei is derived .
    many thanks .
    Exponential Integral -- from Wolfram MathWorld

    It isnt derived it comes from the incapability to get a closed solution to \int\frac{e^{x}}{x}dx
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  8. #8
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    Ei(x)

    I expanded the integrand several terms of \int\frac{e^{x}}{x}dx , and i think i can see why ln (x) is in Ei(x) . Thanks .
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by Demus View Post
    I would like to show symbols . As i am very newbie , i'm in need of tutorials so as to better illustrate my questions .
    We use LaTeX for mathematical typesetting here. You will find the tutorial here

    RonL
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  10. #10
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    Testing 1 2 3 .

    Ei(x) = \gamma + \ln(x) + \sum_{n=1}^{\infty}\frac{x^n}{n\cdot{n!}}

    where \gamma is Euler's constant .

    Thanks for your help guys .
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