  We study initial boundary value problems for linear scalar partial
differential equations with constant coefficients, with spatial derivatives of
{\em arbitrary order}, posed on the domain $\{t>0, 0<x<L\}$. We first show that
by analysing the so-called {\em global relation}, which is an algebraic
relation defined in the complex $k$-plane coupling all boundary values of the
solution, it is possible to identify how many boundary conditions must be
prescribed at each end of the space interval in order for the problem to be
well posed. We then show that the solution can be expressed as an integral in
the complex $k$-plane. This integral is defined in terms of an $x$-transform of
the initial condition and a $t$-transform of the boundary conditions. For
particular cases, such as the case of periodic boundary conditions, or the case
of boundary value problems for {\em second} order PDEs, the integral can be
rewritten as an infinite series. However, there exist initial boundary value
problems for which the only representation is an integral which {\em cannot} be
written as an infinite series. An example of such a problem is provided by the
linearised version of the KdV equation. Thus, contrary to common belief, the
solution of many linear initial boundary value problems on a finite interval
{\em cannot} be expressed in terms of an infinite series.
