  Representation theorems for both isotropic and anisotropic functions are of
prime importance in both theoretical and applied mechanics. The Eshelby
inclusion problem is very fundamental, and is of particular importance in the
design of advanced functional composite materials. In this paper, we discuss
about two-dimensional Eshelby tensors (denoted as $M^{(2)}$). Eshelby tensors
satisfy the minor index symmetry $M_{ijkl}^{(2)}=M_{jikl}^{(2)}=M_{ijlk}^{(2)}$
and have wide applications in many fields of mechanics. In view of the
representation of two-dimensional irreducible tensors in complex field, we
obtain a minimal integrity basis of ten isotropic invariants of $M^{(2)}$.
Remarkably, note that an integrity basis is always a functional basis, we
further confirm that the minimal integrity basis is also an irreducible
function basis of isotropic invariants of $M^{(2)}$.
