  We investigate axisymmetric black hole~(BH) formation and its gravitational
wave (GW) and neutrino signals with self-consistent core-collapse supernova
simulations of a non-rotating $40~M_\odot$ progenitor star using the isotropic
diffusion source approximation for the neutrino transport and a modified
gravitational potential for general relativistic effects. We consider four
different neutron star (NS) equations of state~(EoS): LS220, SFHo,
BHB$\Lambda\phi$ and DD2, and study the impact of the EoS on BH formation
dynamics and GW emission. We find that the BH formation time is sensitive to
the EoS from 460 to $>1300$~ms and is delayed in multiple dimensions for
$\sim~100-250$~ms due to the finite entropy effects. Depending on the EoS, our
simulations show the possibility that shock revival can occur along with the
collapse of the proto-neutron star~(PNS) to a BH. The gravitational waveforms
contain four major features that are similar to previous studies but show
extreme values: (1)~a low frequency signal ($\sim~300-500$~Hz) from core-bounce
and prompt convection, (2)~a strong signal from the PNS g-mode oscillation
among other features, (3)~a high frequency signal from the PNS inner-core
convection, and (4)~signals from the standing accretion shock instability and
convection. The peak frequency at the onset of BH formation reaches to
$\sim~2.3$~kHz. The characteristic amplitude of a 10~kpc object at peak
frequency is detectable but close to the noise threshold of the Advanced~LIGO
and KAGRA, suggesting that the next generation gravitational wave detector will
need to improve the sensitivity at the kHz domain to better observe
stellar-mass BH formation from core-collapse supernovae or failed supernovae.
