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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

    micromass

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    The integral can not be expressed in terms of elementary functions.
     
  4. Oct 22, 2012 #3
  5. Oct 22, 2012 #4

    dextercioby

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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.
     

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  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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