# Gauss integral

• Apr 29th 2008, 02:51 PM
Demus
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.
• Apr 29th 2008, 02:58 PM
Mathstud28
Quote:

Originally Posted by Demus
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

$\sum_{n=0}^{\infty}\frac{x^n}{n\cdot{n!}}$
• Apr 30th 2008, 06:49 AM
CaptainBlack
Quote:

Originally Posted by Mathstud28
$\int_a^{b}\frac{1}{\ln(x)}dx$
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
• Apr 30th 2008, 09:27 AM
Demus
integrals
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 ?
• Apr 30th 2008, 11:40 AM
CaptainBlack
Quote:

Originally Posted by Demus
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):

$
{\rm{Li}}(x)=\int_2^x \frac{1}{\ln(t)}~dt
$

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.

RonL
• Apr 30th 2008, 01:12 PM
Demus
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 .
• Apr 30th 2008, 01:19 PM
Mathstud28
Quote:

Originally Posted by Demus
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 $\int\frac{e^{x}}{x}dx$
• Apr 30th 2008, 03:12 PM
Demus
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 .
• May 1st 2008, 12:32 PM
CaptainBlack
Quote:

Originally Posted by Demus
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
• May 1st 2008, 02:44 PM
Demus
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 .