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

1. Oct 22, 2012

### iVenky

Well I was solving this differential equation and I had to find out the integral of e1/x

$\int e1/x$ dx

Why is this latex thing for integral not working ?

2. Oct 22, 2012

### micromass

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

3. Oct 22, 2012

### JJacquelin

4. Oct 22, 2012

### 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.

Last edited: Oct 22, 2012
5. Oct 23, 2012

### 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.

#### Attached Files:

• ###### Elliptic vs hypergeometric.JPG
File size:
4.7 KB
Views:
144
6. Oct 23, 2012

### 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.