Partial Waves (10-325)

2. Partial waves

Using the expansion of $f(θ)$ in partial waves ℓ and orthogonality, show that the total cross section $σ = R d |f(θ)|^2 d\Omega$ is proportional to $Imf(θ = 0)$.

The scattered wave is

$ψ=f(θ)*(e^{i k r})/r$  where

$f(θ)=∑_{(l=0)}^∞ (2l+1) *a_l P_l (\cos⁡θ)$

where $a_l$ are complex numbers representing the amplitudes of the partial waves.

$dσ/dΩ=|f|^2=∑_m ∑_n (2m+1)(2n+1) a_m^* a_n  P_m (\cos θ) P_n (\cos⁡θ)$


$σ=∫|f|^2 dΩ=4π∑_l (2l+1) |a_l |^2$    since $P_m$  and $P_n$  are orthogonal for $n? m$

We have from the first equation:
$Im (f(θ=0) )=∑_l (2l+1)*Im(a_l )*P_l (1)  =A*∑_l (2l+1)*Im(a_l )$

with A real number

Here I am lost. I do not know how to show that

$|a_l |^2 \sim Im (a_l)$

I guess it is a mistake in text.