Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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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!

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.

Concerning the question when aspects of math are useful:

I've heard/read that question a lot and even though math is an integral part of what I'm doing right now I found that math becomes useful in real life when you yourself make it useful, i.e. you can get around without using math but once you learn more and more aspects of it you start viewing some things differently and may even find that applying concepts from math help you in understanding some things or even let you find a solution more easily.

I realise that this is a bit abstract but you need to get into the mindset to apply that. For example if you have a schedule where you need to make sure that the tasks are evenly distributed among a number of people but some of these people may be available on fewer days you could try to do that with some linear equations etc.

You are right

I edited my post and removed the mistake. So what is the distance between these two parallel lines?

The angle between two n-dimensional vectors u and v can be calculated by using

Suppose we have two vectors

We can use the above formula to calculate the angle.

You have two equations with two variables each.

You can plot these in a two-dimensional coordinate system with the axes x and y.

In this case it would be sensible to rewrite the equations in such a way that you could plot them by plugging in numbers, e.g.

You can see that you have a linear equation. And then you do the same with the other equation and take the difference.

Well, let's say we have two parallel lines

We would figure out whether they are parallel or not by looking for a point where they meet, i.e.

Obviously this is not correct so there is no point where they meet, i.e. they are parallel.

We would calculate the distance by taking the difference between these two lines

**tanith**- Replies: 7

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

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