How Do You Correctly Evaluate an Iterated Integral Involving a Step Function?

[Pietjuh]

For my statistical physics class I need to evaluate the following type of integral by iteration and I am not sure I'm doing it correctly

$$I_N = \int_0^{\infty}dx_1 \int_0^{\infty}dx_2\cdots \int_0^{\infty} \theta(a - \sum_{i=0}^N x_i) dx_N$$

Since i must iterate this integral I'm going to try and calculate the last integral. I then wrote
$$a' = a - \sum_{i=0}^{N-1} x_i$$
so,

$$I_N = \int_0^{\infty}dx_1 \int_0^{\infty}dx_2\cdots \int_0^{\infty} \theta(a' - x_N) dx_N = \int_0^{\infty}dx_1 \int_0^{\infty}dx_2\cdots \int_0^{a'} 1 dx_N = \int_0^{\infty}dx_1 \int_0^{\infty}dx_2\cdots \int_0^{\infty} a' dx_{N-1}$$

But this integral diverges! Can someone tell me what I did wrong here?

 

[member 553083]

You did not do anything wrong. The integral does indeed diverge because the integrand is always equal to 1, and the upper bound of integration does not go to 0 as N goes to infinity. This means that as N increases, the integral increases without limit.

 

[thepatientmental]

It looks like you are on the right track with your approach to evaluating this integral using iteration. However, it seems like there may be a mistake in your final step. Let's take a closer look at the integral you are trying to evaluate:

I_N = \int_0^{\infty}dx_1 \int_0^{\infty}dx_2\cdots \int_0^{\infty} \theta(a - \sum_{i=0}^N x_i) dx_N

In order to evaluate this integral by iteration, we can start by considering the last integral:

J_N = \int_0^{\infty} \theta(a - \sum_{i=0}^N x_i) dx_N

As you correctly pointed out, we can rewrite this as:

J_N = \int_0^{\infty} \theta(a' - x_N) dx_N

where a' = a - \sum_{i=0}^{N-1} x_i. However, in your next step, you seem to have made a mistake by replacing the upper limit of the integral with a'. This is incorrect because the upper limit should still be \infty. Instead, we can rewrite the integral as:

J_N = \int_0^{a'} 1 dx_N

This gives us a finite value for the integral, and we can now continue with the iteration process as you did by replacing x_N with a' in the previous integrals. This should give us a convergent result for the entire integral I_N.

In summary, it seems like the mistake was in replacing the upper limit of the integral with a' instead of keeping it as \infty. I hope this helps clarify the issue and allows you to successfully evaluate the integral.