# (p.360) G An integral

# (p.360) G An integral

The two-dimensional integral in Eq. (11.11) may be reduced to two one-dimensional integrals by a change of variables. This is illustrated in Fig. G.0.1, which shows a Cartesian coordinate system with unit axes (*i,j*) and associated coordinates τ_{1} and τ2.

We now rotate the (*i,j*) system by 45° to the (*i',j'*) system. The orthonormal transformation is given by

**R**represented in the two coordinate systems is

(p.361) Then the relation between the ‘old’ and ‘new’ coordinates is

*t*in Fig. G.0.1. To cover the same area in the new coordinate system, the integral is divided into two integrals, one for the triangle below the line

*l*and one for the triangle above this line. This division is naturally introduced in order to set the limits for ${{\tau}^{\prime}}_{2}$ when we integrate over${{\tau}^{\prime}}_{1}$. For the lower triangle we see from Eqs (G.3) and (G.4) that the region spanned by ${{\tau}^{\prime}}_{2}$ for a given value of ${{\tau}^{\prime}}_{1}$ is

The integral in Eq. (11.11) then becomes

Our assumption that ϕ(θ) is sharply peaked around *θ = 0* and drops to zero for *θ > τ* _{C}implies that its integral *f(z*) reaches a constant value *f* = *f*(∞) for τ ∼ τ_{C}. Therefore,
(p.362)
when we are interested in much longer times, we may replace *f*(ψ) and *f*(2*t − ψ*) by their constant asymptotic value *f* in the right-hand side of Eq. (G.9). Hence we may write, for *t* ≫ τ_{c},

Substitution of this result for the double integral in Eq. (11.11) finally gives us Eq. (11.12).