$\kappa$ were obtained by Harbater-Hartmann-Krashen [@hhk:patching_qf_csa] using patching techniques. This recovers the $u$-invariant of function fields of non-dyadic $p$-adic curves ([@ps:uinv_8]). Leep ([@leep]), using results of Heath-Brown ([@heath-brown]), proved that the $u$-invariant of function fields of all $p$-adic curves (including dyadic curves) is 8. An alternate proof for function fields of dyadic curves is given in ([@ps:period_index]). In fact more generally we proved that if char$(K) = 0$, char$(\kappa) =2$ and $\kappa$ is perfect, then $u(F) \leq 8$. If $[\kappa : \kappa^2]$ is infinite it is easy to construct anisotropic quadratic forms over $K$ and hence over $F$ of arbitrarily large dimension. The question remained open whether the $u$-invariant of $F$ is finite if char$(\kappa) = 2$ and $[\kappa : \kappa^2]$ is finite. The aim of this article is to give an affirmative answer to this question. More precisely we prove the following (\[theorem:u\_invariant\])

Let $K$ be complete discretely valued field with residue field $\kappa$ and $F$ a function field of a curve over $K$. Suppose that char$(K)  = 0$, char$(\kappa) = 2$ and $[\kappa : \kappa^2]$ is finite. Then there exists an integer $M$ which depends only on $[\kappa :
\kappa^2]$ such that for any finite extension $F$ of 