  We consider the Cauchy problem for the complex valued semi-linear heat
equation $$
  \partial_t u - \Delta u - u^m =0, \ \ u (0,x) = u_0(x), $$ where $m\geq 2$ is
an integer and the initial data belong to super-critical spaces $E^s_\sigma$
for which the norms are defined by $$ \|f\|_{E^s_\sigma} = \|\langle
\xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ \ \sigma \in \mathbb{R}, \
s<0. $$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s_\sigma$,
i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s_\sigma$. We obtain the global
existence and uniqueness of the solutions if the initial data belong to
$E^s_\sigma$ ($s<0, \ \sigma \geq d/2-2/(m-1)$) and their Fourier transforms
are supported in the first octant, the smallness conditions on the initial data
in $E^s_\sigma$ are not required for the global solutions. Moreover, we show
that the error between the solution $u$ and the iteration solution $u^{(j)}$ is
$C^j/(j\,!)^2$. Similar results also hold if the nonlinearity $u^m$ is replaced
by an exponential function $e^u-1$.
