Definite exponential integral with argument in square root

tanith · 2017-04-26 00:29:57

Hi,

I'm really struggling with this definite integral.

Unfortunately I cannot say whether this integral even has a solution. I was told it might be somehow related to Bessel functions but frankly I do not know.
I've tried substitution but this does not lead me to a form that is solvable.
Looking through lists with integrals didn't get me anywhere close to the solution either.

So I'm asking because I'm keen to know whether this integral even has an analytic solution and how to obtain it.

Cheers

Last edited by tanith (2017-04-26 03:06:05)

bobbym · 2017-04-26 07:37:59

Hi;

Can you say something about a and b?

tanith · 2017-04-26 19:04:09

Hey, I think the only useful thing that I can say is that a and b are real numbers, positive numbers and are not strictly bound.

bobbym · 2017-04-27 01:31:31

Hi;

How did this integral come to be? Research project? Contest problem? Book problem?

tanith · 2017-04-27 21:05:56

This is a partition function from thermodynamics.
Partition function (wikipedia)
It's the formula under Classical discrete systems, however, it's written a bit differently, since in my case it's a continuous function.

I'm currently working on my PhD in physics and I've always evaluated this integral numerically. I was just curious whether there is an actual analytic solution to this function.

The formula is:

I think it's usually found in this form:

So when you do the following transformation

and use the relativistic energy-momentum-mass equality in natural units, i.e.

you get

It's a well-behaved integral, without any singularities or oscillations or something like that so a numerical evaluation is pretty straightforward, however, it's a bit annoying since I have to evaluate it every time I change the temperature.

bobbym · 2017-04-27 23:06:50

Hi;

Even when I substitute numbers for both m and T I still am unable to get a closed form.

The curve does look like a probability density function though.

tanith · 2017-04-27 23:22:26

You are right! It's something like a probability density function. If you fold it with another probability function for some specific particle you can get a mass distribution of that particle in a nuclear medium.

My PI gave me an old mathematics book but I wasn't able to find that integral there. Neither could I find it online or in other forums so this leads me to believe that the solution is not known, at least not to me and as far as I could look for it. As a last resort I could probably go to the other campus and ask some colleagues from the mathematics department but I'm not sure whether this will do any good.

I thank you for your time though!

bobbym · 2017-04-28 02:56:24

Hi;

If you get any solution please post it here.

Math Is Fun Forum

#1 2017-04-26 00:29:57

Definite exponential integral with argument in square root

#2 2017-04-26 07:37:59

Re: Definite exponential integral with argument in square root

#3 2017-04-26 19:04:09

Re: Definite exponential integral with argument in square root

#4 2017-04-27 01:31:31

Re: Definite exponential integral with argument in square root

#5 2017-04-27 21:05:56

Re: Definite exponential integral with argument in square root

#6 2017-04-27 23:06:50

Re: Definite exponential integral with argument in square root

#7 2017-04-27 23:22:26

Re: Definite exponential integral with argument in square root

#8 2017-04-28 02:56:24

Re: Definite exponential integral with argument in square root

Board footer