# Small phase approximation

Does anyone known how to prove that if $$|\phi_k (r)| \ll 1$$ for all $$r$$ and all $$k=1,...,n\,$$, the following equation $$S=\left|\int_0^\infty A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2$$ can be simplified to $$\left(\frac{S}{S_0}-1\right) \approx \sum_{k=1}^n\left(\frac{S_k}{S_0}-1\right),$$ where $$S_0=\left|\int_0^\infty A(r)e^{-i\phi_0(r)} dr \right|^2$$ and $$S_k=\left|\int_0^\infty A(r)e^{-i[\phi_0(r)+\phi_k(r)]} dr \right|^2 \,\,\,\,\, ?$$

Additionally, $$A(r)$$ and $$\phi_0(r)$$ are functions having non-negative real values for all $$r$$, $$\phi_k(r)$$ are small phase functions ($$|\phi_k (r)| \ll 1$$) with real values for all $$r$$ and all $$k=1,...,n\,$$, and $$\lim_{r\to\infty}⁡A(r)=0$$.

Example functions: $$A(r)=(2r/r_0^2)e^{-r^2/r_0^2}$$ and $$\phi_0 (r)=Br^2/r_0^2$$, where $$r_0$$ and $$B$$ are positive real numbers.

Linearization in $$\phi_k(r)$$ gives the desired approximation: $$S=\left|\int_0^\infty dr\, A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2$$ $$=\int_0^\infty dr\,\int_0^\infty dr'\, A(r)A(r')e^{i\phi_0(r')-i\phi_0(r)}\left(1+i\sum_{k=1}^n [\phi_k(r')-\phi_k(r)]+{\cal O}(\phi^2) \right)$$ $$=S_0+\sum_{k=1}^n\int_0^\infty dr\,\int_0^\infty dr'\, A(r)A(r')e^{i\phi_0(r')-i\phi_0(r)}\left[e^{i[\phi_k(r')-\phi_k(r)]}-1+{\cal O}(\phi^2)\right]$$ $$=S_0+\sum_{k=1}^N (S_k-S_0)+{\cal O}(\phi^2).$$