Many applications require a computer representation of 2D shape, usually described by a set of 2D points. The challenge of this representation is that it must not only capture the characteristics of the shape but also be invariant to relevant transformations. Invariance to geometric transformations, such as translation, rotation and scale, has received attention in the past, usually under the assumption that the points are previously labeled, i.e., that the shape is characterized by an ordered set of landmarks. However, in many practical scenarios the landmarks are obtained from an automatic process, e.g., edge/corner detection, thus without natural ordering. In this paper, we represent 2D shapes in a way that is invariant to the permutation of the landmarks. Within our framework, a shape is mapped to an analytic function on the complex plane, leading to what we call its analytic signature (ANSIG). We show that different shapes lead to different ANSIGs but that shapes that differ by a permutation of the landmarks lead to the same ANSIG, i.e., that our representation is a maximal invariant with respect to the permutation group. To store an ANSIG, it suffices to sample it along a closed contour in the complex plane. We further show how easy it is to factor out geometric transformations when comparing shapes using the ANSIG representation. We illustrate the ANSIG capabilities in shape-based image classification.