  We study the time and temperature dependent correlation functions for an
impenetrable Bose gas with Neumann or Dirichlet boundary conditions $\langle
\psi(x_1,0)\psi^\dagger(x_2,t)\rangle _{\pm,T}$. We derive the Fredholm
determinant formulae for the correlation functions, by means of the Bethe
Ansatz. For the special case $x_1=0$, we express correlation functions with
Neumann boundary conditions $\langle\psi(0,0)\psi^\dagger(x_2,t)\rangle
_{+,T}$, in terms of solutions of nonlinear partial differential equations
which were introduced in \cite{kojima:Sl} as a generalization of the nonlinear
Schr\"odinger equations. We generalize the Fredholm minor determinant formulae
of ground state correlation functions $\langle\psi(x_1)\psi^\dagger(x_2)\rangle
_{\pm,0}$ in \cite{kojima:K}, to the Fredholm determinant formulae for the time
and temperature dependent correlation functions
$\langle\psi(x_1,0)\psi^\dagger(x_2,t)\rangle _{\pm,T}$, $t \in {\bf R}$, $T
\geq 0$.
