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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.

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

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