  A systematic method for the computation of finite temperature ($T$) crossover
functions near quantum critical points close to, or above, their upper-critical
dimension is devised. We describe the physics of the various regions in the $T$
and critical tuning parameter ($t$) plane. The quantum critical point is at
$T=0$, $t=0$, and in many cases there is a line of finite temperature
transitions at $T = T_c (t)$, $t < 0$ with $T_c (0) = 0$. For the relativistic,
$n$-component $\phi^4$ continuum quantum field theory (which describes lattice
quantum rotor ($n \geq 2$) and transverse field Ising ($n=1$) models) the upper
critical dimension is $d=3$, and for $d<3$, $\epsilon=3-d$ is the control
parameter over the entire phase diagram. In the region $|T - T_c (t)| \ll T_c
(t)$, we obtain an $\epsilon$ expansion for coupling constants which then are
input as arguments of known {\em classical, tricritical,} crossover functions.
In the high $T$ region of the continuum theory, an expansion in integer powers
of $\sqrt{\epsilon}$, modulo powers of $\ln \epsilon$, holds for all
thermodynamic observables, static correlators, and dynamic properties at all
Matsubara frequencies; for the imaginary part of correlators at real
frequencies ($\omega$), the perturbative $\sqrt{\epsilon}$ expansion describes
quantum relaxation at $\hbar \omega \sim k_B T$ or larger, but fails for $\hbar
\omega \sim \sqrt{\epsilon} k_B T$ or smaller. An important principle,
underlying the whole calculation, is the analyticity of all observables as
functions of $t$ at $t=0$, for $T>0$; indeed, analytic continuation in $t$ is
used to obtain results in a portion of the phase diagram. Our method also
applies to a large class of other quantum critical points and their associated
continuum quantum field theories.
