Integration bounds after change of variables

[dand5]

I have a quick question about integration after a change of variables has been made.

Suppose there is a function $$R(t_{1},t_{2})$$ that actually just
depends on the difference $$t_{1} - t_{2}$$. The goal is then to
simplify the following integral:

$$\frac{1}{T^{2}}\int^{T}_{0}\int^{T}_{0} R(t_{1},t_{2}) dt_{1}dt_{2}$$

by using the substitution $$t_{1}' = t_{1}$$ and $$t_{2}'= t_{1} - \tau$$.

A straight substitution yields:
$$\frac{1}{T^{2}}\int \int^{T}_{0} R(\tau) dt_{1}'(dt_{1}' - d\tau)$$

I am uncertain about two things:

1) the integration bounds on the outer integral after the substitution has been made
2) whether or not $$dt_{1}'$$ in the outer integral is zero since
$$dt_{1}$$ is held constant when integrating over $$dt_{2}$$ before the substitution was made.

As a heads up the final result is supposed to be:
$$\frac{1}{T^{2}}\int^{T}_{-T}\left(T-\left|\tau\right|\right)R(\tau) d\tau$$

Thanks in advance for any responses.

 

[mathman]

To simplify writing I will use s instead of tau.

Let s=t1-t2, t=t2. Then the differential looks like R(s)dsdt, with s limits -t,T-t and the t limits 0,T.

Next reverse the order of integration. The t limit is split into 2 parts. For s<0, the limits are -s and T, while for s>0, the limits are 0 and T-s. The s limits are -T and T.

When you integrate with respect to t, you get T-|s|.

 

[dand5]

ok I see it now, Thanks. Also, the determinant of the Jacobian of the coordinate transformation is 1, so I guess it is correct to say that dt1dt2=dsdt.