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For a time Laplace transform parameter with $\mathrm{Re}p>0$, the spatial characteristic roots satisfy $r^3+r+p=0$. Exactly one root $r_{-}$ has negative real part and two have positive real part. Indeed an imaginary root would force $p=-i(k-k^3)$ to be imaginary. The count is therefore constant on the connected right half-plane; for real $p>0$, the cubic has one negative real root and a conjugate pair with positive real parts. Repeated roots would satisfy $3r^2+1=0$, which again puts $p$ on the imaginary axis. Thus $r_{-}(p)$ is a nonzero holomorphic function in the right half-plane. Only $e^{r_{-}x}$ decays at positive spatial infinity.

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.

The structure sheaf assigns to an open set its ring of holomorphic functions. Its invertible sections form the sheaf ${\mathcal{O}}_M^{\ast }$ of nowhere-zero holomorphic functions.

Here a weak modular form is a holomorphic function on the complex upper half-plane satisfying the weight-$k$ transformation law, with no growth condition at any cusp of a modular group. This convention differs from weakly holomorphic modular forms, which are required to be meromorphic at the cusps. At level one, translation invariance gives a Laurent expansion on the punctured unit disc, possibly with infinitely many negative powers.

Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.