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What is the integral of e^(1/x)

  1. Oct 22, 2012 #1
    Well I was solving this differential equation and I had to find out the integral of e1/x

    [itex]\int e1/x[/itex] dx

    Thanks in advance.

    Why is this latex thing for integral not working ?
  2. jcsd
  3. Oct 22, 2012 #2
    The integral can not be expressed in terms of elementary functions.
  4. Oct 22, 2012 #3
  5. Oct 22, 2012 #4


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    Can you post the ODE, you might have done a mistake somewhere.

    @Jean: Do you know if there's a connection (functional relation) between certain hypergeometric functions and the complete/incomplete elliptic integrals ? I suspect there might be one.
    Last edited: Oct 22, 2012
  6. Oct 23, 2012 #5
    @ dextercioby:

    The relationships between Complete Elliptic Integrals E(x), K(x) and Gauss Hypergeometric functions are shown in attachment.
    I don't know about such relationship for Incomplete Elliptic Integrals. I suppose that it would be much more complicated to develop those integrals into hypergeometric series. If possible, most likely this would involve hypergeometic functions of higher level than 2F1.

    Attached Files:

  7. Oct 23, 2012 #6
    If all you need is *an answer* then...

    step 1: expand e^x into a power series: e^x = 1 + x + (1/2!)x^2 + (1/3!)x^3 + ...
    step 2: substitute 1/x for x: e^(1/x) = 1 + x^-1 + (1/2!)x^-2 + (1/3!)x^-3 + ...
    step 3: integrate each term of the power series: x + ln x -(1/2!)x^-1 - (1/2)(1/3!)x^-2 +...

    if i made an algebra mistake, sorry... but the idea is clear.
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