exponential integral

abotaha · 2010-07-18 23:24:31

Dear all,
I have exponential integral with polynomial. I tried to solve it but I could not.
the integral is :

I complete the square of the exponential power and it looks like:

but it is still not easy for me to solve it

Can any one help me please.

Thanks in advance.

bobbym · 2010-07-19 00:16:47

Hi abotaha;

It is not easy for anyone to do that. The answer involves the error function which is not an elementary function. It is unlikely that it can be done by hand. As a matter of fact wolfram alpha is having problems with it. If you need an answer I can get something for you.

abotaha · 2010-07-19 05:50:05

Thanks bobbym for the response.
I have did the following simplification and the real bound value of the integration is from 0 to infinity:

and I separate that into three integrals:

Right now the integration seems to be simply, but it is still not easy for me to do it.
I need help please .

Last edited by abotaha (2010-07-19 05:50:40)

bobbym · 2010-07-19 06:58:37

Hi;

I will do them one at a time but remember, two of the terms will involve the erf function. We replace infinity by t.

Provided what you gave me is correct that is the answer. But you are not out of trouble by a long shot. You must now define m for one thing before we can continue.

abotaha · 2010-07-19 07:20:18

It is nice to have this solution. I wonder if there is any way to avoid erf function.

bobbym · 2010-07-19 07:26:07

Hi;

Hold it, we are not done, not even close. You must bound m and t. This is important!

abotaha · 2010-07-19 22:13:30

Hi guys,

The above integrals can be expressed in terms of the Euler gamma function:

and suppose

then

Then, use the rules:

the problem now is that the boundaries of the integral after some transformation steps are changed such that start from constant to infinity as:

so when i expressed the last integral in terms of the Euler gamma function I got:

observe the lower integral boundary is constant (c).
is there any transformation to make the low boundary zero in order to apply Gamma function formula, or is there any method to deal with this case?

bobbym · 2010-07-20 02:23:01

Hi abotaha;

Yes, I can translate it to 0, infinity. But c isn't even in the RHS.

My confusion stems from the fact that every time I get close to some solution, you steer the problem in a new direction. I love looking at new ideas but having 3 different incomplete answers is only going to result in never getting an answer. I thought we were making headway with post #4. Let's try to finish one method before we move on to another one. We still don't klnow whether any of these methods are going to get the right answers. Also restructuring in terms of the gamma function is no easier than in terms of the error function. Both are tabulated functions and we can worry about them at the end. So, what one do you like?

abotaha · 2010-07-20 07:49:52

bobbym wrote:

Hi abotaha;
Yes, I can translate it to 0, infinity. But c isn't even in the RHS.
My confusion stems from the fact that every time I get close to some solution, you steer the problem in a new direction. I love looking at new ideas but having 3 different incomplete answers is only going to result in never getting an answer. I thought we were making headway with post #4. Let's try to finish one method before we move on to another one. We still don't klnow whether any of these methods are going to get the right answers. Also restructuring in terms of the gamma function is no easier than in terms of the error function. Both are tabulated functions and we can worry about them at the end. So, what one do you like?

Sorry for multi and different way of expressing the questions. I also try to time after time to simplify the problem and that way i am looking for many way in order to solve the integral in general ( analytically) without special case.
post #4 was a good solution, however i do not like to user erf function because it does not support an exact solution of my work.
In this case it is better to include gamma function as a solving tool.

if you could please help me with the gamma function, since i found a way to deal with the boundaries integral like what i have, it is incomplete gamma function:

where

;

This is only work when n is an integer, however, this is not the case that i am considering since n is not integer in my problem.

Any suggestion please.

bobbym · 2010-07-20 15:39:50

Hi;

I don't understand what you need done. You need a form for:

When n is not an integer?

Also, you have some misconceptions about integration. You are stating that the integral has been analytically done, because it is in terms of the gamma function. The gamma function is just a name given to another integral. When you have removed all integral signs then you have analytically done the integral. Not when you have replaced it by yet another integral.

however i do not like to user erf function because it does not support an exact solution of my work.

The gamma function for non integer values is not easy to compute. It is a special case, and the incomplete gamma function is going to require more work. It is not an exact solution.

In this case it is better to include gamma function as a solving tool.

Here is why the erf is very much superior to the gamma function here . Sooner or later since there is probably no analytical answer to your integrand you are going to have to resort to numerical methods ( if you actually want to compute anything with your answer).

Remember when I said it was now time to bound t? We will have to sooner or later truncate that infinity to some reasonable number. Check out the properties of gamma(t) versus erf(t). When t approaches infinity so does gamma(t). No truncation is possible, you are going have to work with the infinity in some quadrature idea. Notice when t approaches infinity, erf(t) approaches 1. That means it will be possible to get as much accuracy as we require saying that t = 10 or t= 20... We can replace infinity with some number. No matter how large that number may be it will still be a number!

I am not saying the above idea will work for sure, but it has a chance!

abotaha · 2010-07-20 22:32:02

Hi Bobbym,
thanks for this long explanation.
based on what i understood and i realized that it is better to use the solution that you suggested (post #4).

It is really helps a lot and i noticed that especially after your explanation.

thanks again.

abotaha.

bobbym · 2010-07-21 03:15:53

Hi abotaha;

We will probably have to try both of them. What do you need done?

George,Y · 2010-07-21 18:48:26

if you can manipulate the integral into the form

then it is solvable

parameter h is not important
you need to get

(-a,b,-c) = (-a,m+n,-c)
and
(-a,m) = p(-d,f); (b/2,-c) = q(-d,f)

Last edited by George,Y (2010-07-21 18:48:53)

George,Y · 2010-07-21 18:50:20

wait, it is still not solvable for the p part...

Math Is Fun Forum

#1 2010-07-18 23:24:31

exponential integral

#2 2010-07-19 00:16:47

Re: exponential integral

#3 2010-07-19 05:50:05

Re: exponential integral

#4 2010-07-19 06:58:37

Re: exponential integral

#5 2010-07-19 07:20:18

Re: exponential integral

#6 2010-07-19 07:26:07

Re: exponential integral

#7 2010-07-19 22:13:30

Re: exponential integral

#8 2010-07-20 02:23:01

Re: exponential integral

#9 2010-07-20 07:49:52

Re: exponential integral

#10 2010-07-20 15:39:50

Re: exponential integral

#11 2010-07-20 22:32:02

Re: exponential integral

#12 2010-07-21 03:15:53

Re: exponential integral

#13 2010-07-21 18:48:26

Re: exponential integral

#14 2010-07-21 18:50:20

Re: exponential integral

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