  We study the pole properties of $\Lambda(1405)$ in a model-independent manner
by applying the Uniformized Mittag-Leffler expansion proposed in our previous
paper. The resonant energy, width and residues are determined by expanding the
observable as a sum of resonant-pole pairs under an appropriate
parameterization which expresses the observable to be single-valued, and
fitting it to experimental data of the invariant-mass distribution of
$\pi^+\Sigma^-$, $\pi^-\Sigma^+$, $\pi^0\Sigma^0$ final states in the reaction,
$\gamma p \rightarrow K^+ \pi \Sigma$, and the elastic and inelastic cross
section, $K^-p\to K^-p$, $\bar{K}^0n$, $\pi^+\Sigma^-$, $\pi^-\Sigma^+$. As we
gradually increase the number of pairs from one to three, the first pair
converges while the second and third pairs emerge further and further away from
the first pair, implying that the Uniformized Mittag-Leffler expansion with
three pairs is almost convergent in the vicinity of the $\Lambda(1405)$. The
broad peak structure between the $\pi\Sigma$ and $\bar{K} N$ thresholds
regarded to be $\Lambda(1405)$ is explained by a single pair with a resonant
energy of 1420 $\pm$ 1 MeV, and a half width of 48 $\pm$ 2 MeV, which is
consistent with the single-pole picture of $\Lambda(1405)$. We conclude that
the Uniformized Mittag-Leffler expansion turns out to be a very powerful method
to obtain resonance energy, width and residues from the near-threshold
spectrum.
