# Solving the Integral for Boltzmann Law for Scientists

• jaap de vries
In summary: Does anybody have any idea on how to do this elegantly??Thank you.In summary, Jaap found a problem with integrating the fomula to obatin boltzman law. He is looking for someone to help him with the solution.
jaap de vries
When integrating plank's fomula to obatin boltzman law,
I need to integrate

f(x) = x^3/(e^x-1) from 0 to infinity, the result is pi^4/15

Does anybody have any idea on how to do this elegantly??
Thank you.
Jaap

This is a problem in Arfken. It involves the ploygamma function and the Riemann zeta function. I won't have time to think about it until later today.

Regards,
George

jaap de vries said:
When integrating plank's fomula to obatin boltzman law,
I need to integrate
f(x) = x^3/(e^x-1) from 0 to infinity, the result is pi^4/15
Does anybody have any idea on how to do this elegantly??
Thank you.
Jaap

This is a problem previously addressed by Daniel:

The integral is the Debye-Einstein integral:

$$\mathcal{D}_3=\int_0^{\infty} \frac{x^3}{e^x-1}dx=\int_0^\infty \frac{x^3e^{-x}}{1-e^{-x}}dx$$

Since:

$$\frac{1}{1-e^{-x}}=\sum_{n=0}^{\infty} \left(\frac{1}{e^x}\right)^n$$

then:

$$\sum_{n=1}^{\infty}\int_0^{\infty} x^3 e^{-nx}dx=\Gamma(x)\zeta(4)$$

Last edited:
saltydog said:
This is a problem previously addressed by Daniel

Very nice.

Picking a nit - there's a minor typo in the last line.

I had hoped to have a go at this problem this afternoon after finishing my "real" work; now I guess I'll have to find something else to do.

Regards,
George

George Jones said:
Very nice.
Picking a nit - there's a minor typo in the last line.
Regards,
George

Thanks for pointing that out. Should it read:

$$\sum_{n=1}^{\infty}\int_0^{\infty} x^3 e^{-nx}dx=\Gamma(4)\zeta(4)$$

And thus, would we have:

$$\mathcal{D}_n=\Gamma(n+1)\zeta(n+1)\quad ?$$

I'm not sure and will need to look at it a bit. Well, . . . how about you Jaap?

Edit:

Yep, yep, I think we should re-phrase the question:

Japp, kindly prove or disprove the following:

$$\int_0^{\infty}\frac{x^n}{e^x-1}dx \:?=\:\Gamma(n+1)\zeta(n+1)$$

(and he also showed me how to put that question mark on top of the equal sign but I forgot)

Last edited:
Thanks guys! let me chew on that one a bit. Note however, I'm an engineer not a mathematician. Nice to know there is a community out here to help, Makes me feel good.

I'll let Y'all know if I have any questions

Jaap

## 1. How do I know which integration method to use?

There are several integration methods, such as substitution, integration by parts, and trigonometric substitution. The best way to determine which method to use is to look at the integrand and see if it can be rewritten in a simpler form using one of these methods.

## 2. Can I solve an integral without using integration methods?

There are some integrals that can be solved without using integration methods, such as those with simple and well-known antiderivatives. However, for most integrals, integration methods are necessary to find the solution.

## 3. How do I handle improper integrals?

Improper integrals are those with infinite limits of integration or with discontinuities in the integrand. To solve these integrals, you may need to use techniques such as limits, integration by parts, or trigonometric substitution.

## 4. Can I use a calculator to solve integrals?

Some calculators have built-in integration functions that can solve basic integrals. However, it is important to understand the concepts and methods behind integration in order to use these functions correctly.

## 5. Is there a shortcut to solving integrals?

There are some integrals that have well-known antiderivatives and can be solved quickly. However, for most integrals, there is no shortcut and the proper integration method must be used to find the solution.

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