# exponential integral

Discussion in 'Math' started by ihaveaquestion, Nov 15, 2009.

1. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
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Hi everyone,

I need to find out what (using an integral table) the integral of

e^(-x/a)^2 from negative inf to positive inf is, but I'm having trouble finding it... can somebody help me please and also provide a source?

thanks

Mar 6, 2009
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3. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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Yes that's the one I'm looking for however evaluated from minus inf to positive inf, in the link you provided it says "erf" for error though

4. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
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I think erf(±∞)=±1.0

So you should be able to evaluate.

5. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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What do I evaluate from -1 to 1?

6. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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Also,

Is it possible to just evaluate the integral manually? factor our the 1/e^(a^2) and evaluate 1/e^(x^2) (which you'll then need an integral table for but wolfram integral solver gives the integral of 1/e^(x^2) as sinh(x) - cosh(x)?

7. ### steveb Senior Member

Jul 3, 2008
2,433
469
No that won't work. You can transform the integral with a variable substitution. For example u=x^2/a^2. This will change the integral you need to evaluate to something like e^u/sqrt(u). However, I believe that any form you come up with will not be possible to evaluate by anything other than a series solution. This is why people use the "erf" function to symbolize the answer to integrals of that function for any integral limits. This function is used so often that it can be considered the closed form answer, much as we do with sine, cosine, Bessel and other functions.

The integral from minus infinity to plus infinity should show up in a table of definite integrals, rather than a table for indefinite integrals. For example, in the following link.

http://en.wikipedia.org/wiki/Table_of_integrals#Table_of_Integrals

By the way. I think you wrote the function wrong before. Did you mean -(x/a)^2 rather than (-x/a)^2?

8. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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This is a little frustrating... is this not a common integral? the author in this book says "according to my integral table" and he evaluates an integral very similar to the one I'm asking about...

I'm trying to find the integral of (A*e^-x/a)^2 where A and a are constants from negative infinity to positive infinity

Jul 7, 2009
1,585
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You need to write your integrand with more care -- you've given two different forms. Your first was

$e^{(-x/a)^2}$

It's easy to see that the integral of this from -∞ to +∞ will diverge.

$(Ae^{-x/a})^2$

which is proportional to e^(-bx) where b is a constant -- this is easy to integrate.

I suspect the integrand you actually have is proportional to

$e^{-x^2}$

This integrand is proportional to the famous "bell curve" and is not integrable in closed form, which is why you can't find it in a table of integrals. As someone else mentioned, it's related to the error function, commonly abbreviated as erf(x).

If you're looking for the evaluation of

$\int_{-\infty}^{\infty} e^{-x^2} dx$

it's done with the help of converting it to a double integral in polar coordinates (a clever technique probably due to a genius like Euler). See e.g. Kreyszig, "Advanced Engineering Mathematics", 2nd ed., 1967, pg 713.

I suggest you learn how to use the LaTeX equation-writing features of this board -- they're easy to learn and convenient. It makes it easier to understand mathematical expressions -- and makes it easier to see your typos.

Unlike text messaging on phones, you have to learn to be careful and precise in your statement of math problems, as a tiny typo or mistake can change the problem completely.

10. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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Last edited: Nov 15, 2009
11. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
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Unless the author cited the book from which he unearthed the particular integral, then you have no way of knowing what he is referring to. He's probably not talking about a high school or college text book - this would more likely be something a 'serious' mathematician would have in their library or at least have ready access to.

I'm no 'serious' mathematician by any stretch of the imagination, but I have used books which were essentially large volumes containing extensive tables of integrals. I have no doubt that if I could lay my hands on one of them again I would find a similar integral in a table.

BTW - did you try the Wolfram solution for your integral ...

http://integrals.wolfram.com/index.jsp?expr=(exp(-x^2/a^2))^2&random=false

12. ### steveb Senior Member

Jul 3, 2008
2,433
469
You seem to be ignoring all the good advice you've been given. The question has been answered several times over. You need to thoroughly look at the information you've been given here and available on the net. For example, the reference I gave you shows you the integral in the table of "DEFINITE" integrals. Definite integrals are integrals which include the fixed limits and which only have a constant value as an answer.

13. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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I asked tnk how to evaluate that erf stuff... not sure.

Also, steveb, are you referring to the Gaussian integral on the wiki link?

I think I also now want to evaluate the integral from 0 to some constant, not infinity

14. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
You probably need to review your integration skills.

Here is the solution - based on erf(±∞)=±1

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15. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
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Why do I get a different answer by using a substitution u = -2x/a?

16. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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I know why... because you're solving the integral in the link from the book... I'm solving the integral that I mentioned earlier... (Ae^-x/a)^2

17. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
Hello ihaveaquestion

You seem to be wandering around various problems in this thread without any apparent consistent objective.

If you could carefully state the exact form of the problem to be solved we might make some progress.

Otherwise folks will just lose interest and that's probably not what you want.

Rgds,

t_n_k

18. ### ihaveaquestion Thread Starter Active Member

May 1, 2009
314
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Sorry about that everyone... let's start over.

I'm trying to find the integral of (e^(-x/a))^2 from 0 to positive infinity.

here it is in handwriting to avoid confusion
http://img43.imageshack.us/img43/5332/50060496.jpg

I believe the substitution u= -2x/a can be used but i get confused when evaluating the boundaries

May 1, 2009
314
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20. ### steveb Senior Member

Jul 3, 2008
2,433
469
You're going to kick yourself when you see how easy that one is.

If you want to use substitution, use u=e^(-x/a). But, that is needlessly complicated.

However, the straightforward way to do is is to realize that squaring the exponential is trivial as follows:

(e^(-x/a))^2=e^(-2x/a)

Then, you just use simple integration of an exponential function.