  The tomographic Alcock-Paczynski (AP) method is a promising method that uses
the redshift evolution of the anisotropic clustering in redshift space to
calibrate cosmology. It extends the applicable range of AP method to
substantially nonlinear scales, yielding very tight cosmological constraints.
For future stage-IV slitless spectroscopic surveys, the non-negligible redshift
errors might reduce the advantage of the tomographic AP method by suppressing
the resolution of the nonlinear structure along the line of sight. The present
work studies how redshift errors propagate to cosmological parameters in the
tomographic AP analysis. We use a formula $\sigma_z = \sigma(1+z)^{\alpha} $ to
model the redshift errors, with $\sigma$ varying from 0.001 to 0.006 and
$\alpha$ varying from 0.5 to 1.5. The redshift errors produce a signal of
anisotropic clustering that is similar to a strong finger-of-god effect, which
smears out both the AP signal and the contamination caused by the redshift
space distortions (RSD). For the target precision of the Chinese Space Station
Telescope optical survey ($\sigma\lesssim 0.002$), the decrement of
constraining power on the dark energy equation of state is mild ($\lesssim
50\%$), and the suppression of RSD contamination leads to a smaller
bias-to-signal ratio. Our results indicate that the tomographic AP method will
remain a useful and complementary tool for analyses of future slitless
spectroscopic surveys.
