  We develop an asymptotic analysis of target fluxes in the three-dimensional
(3D) narrow capture problem. The latter concerns a diffusive search process in
which the targets are much smaller than the size of the search domain. The
small target assumption allows us to use matched asymptotic expansions and
Green's functions to solve the diffusion equation in Laplace space. In
particular, we derive an asymptotic expansion of the Laplace transformed flux
into each target in powers of the non-dimensionalized target size $\epsilon$.
One major advantage of working directly with fluxes is that one can generate
statistical quantities such as splitting probabilities and conditional first
passage time moments without having to solve a separate boundary value problem
in each case. However, in order to derive asymptotic expansions of these
quantities, it is necessary to eliminate Green's function singularities that
arise in the limit $s\rightarrow 0$, where $s$ is the Laplace variable. We
achieve this by considering a triple expansion in $\epsilon$, $s$ and
$\Lambda\sim \epsilon /s$. This allows us to perform partial summations over
infinite power series in $\Lambda$, which leads to multiplicative factors of
the form $\Lambda^n/(1+\Lambda)^n $. Since $\Lambda^n/(1+\Lambda)^n \rightarrow
1$ as $s\rightarrow 0$, the singularities in $s$ are eliminated. We then show
how corresponding asymptotic expansions of the splitting probabilities and
conditional MFPTs can be derived in the small-$s$ limit. Finally, we illustrate
the theory by considering a pair of targets in a spherical search domain, for
which the Green's functions can be calculated explicitly.
