How to Integrate e^(1/x)?

[iVenky]

Well I was solving this differential equation and I had to find out the integral of e^1/x


$\int e^1/x$ dx

Thanks in advance.

Why is this latex thing for integral not working ?

 

[micromass]

The integral can not be expressed in terms of elementary functions.

 

[JJacquelin]

Hi !

The integral of exp(1/x) cannot be expressed with the combination of a finite number of elementary functions. It requires the use of a special fuction :
http://www.wolframalpha.com/input/?i=integrate+exp(1/x)&x=0&y=0
About the use of special fuctions, a review paper : "Safari in the contry of Special Functions" :
http://www.scribd.com/JJacquelin/documents
The needed special function Ei is mentioned in this article, on Table 4, p32

 

[dextercioby]

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.

 

[JJacquelin]

@ 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.

 

[chill_factor]

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.