# Product of complex conjugate functions with infinite sums

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) ,$$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and ##a_l## is a the complex term (known as partial wave amplitude). Then, the author calculates the product of ##f(\theta)## with its complex conjugate, and shows the result: $$f(\theta)f^{*}(\theta) = |f(\theta)|^2 = \sum_{l}\sum_{l'}(2l+1)(2l'+1)(a_l)^{*}(a_{l'})P_l(cos\theta)P_{l'}(cos\theta).$$ My problem here is to understand how he obtained the second equation. I'm not familiar with operations with sums and real analysis, and I'm stuck with it. Any help will be very appreciated.

mfb
Mentor
Which part is unclear? The second equation directly follows from the first. The sums work like with real numbers. The complex conjugation is only relevant for the complex amplitude, where you also find it in the second equation.