Let \(X\) be a smooth projective curve over a perfect field \(k\). Let \(W(k(X),\Omega_{k(X)})\) be the Witt group of the function field of \(X\) with values in the modules of differentials \(\Omega_{k(X)}\) of \(k(X)\). A residue homomorphism \(\partial\colon W(k(X),\Omega_{k(X)})\to \bigoplus_{x\in X} W(k(x))\) is defined in [\textit{W.-D. Geyer}, \textit{G. Harder}, \textit{M. Knebusch} and \textit{W. Scharlau}, Invent. Math. 11, 319--328 (1970; Zbl 0203.53401)]. In this paper, a description of the kernel and cokernel of \(\partial\) for smooth hyperelliptic curves with a rational point of ramification over \(\mathbb{P}^ 1\) is given. The kernel of \(\partial\) is the Witt group of \(X\). To describe \(\text{Coker}(\partial)\), the authors introduce a ramified and an unramified part of the cokernel. Under the rationality condition \(_4\text{Pic}(X)={} _4\text{Pic}(X_{\bar k})\), \(\bar k\) denoting the algebraic closure of \(k\), they show that the ramified part is zero and that the unramified part can be described in terms of certain cohomology groups. If, furthermore, \(X\) is defined over a local field with residue field of characteristic not 2 and if \(X\) has good reduction, this description is used to prove that \(\text{Coker}(\partial)\simeq W(k)\oplus(\mathbb{Z}/2)^{4g}\), where \(g\) is the genus of the curve. Finally the authors show that, under the same hypothesis, \(W(X)\) is also isomorphic to \(W(k)\oplus(\mathbb{Z}/2)^{4g}\).