Index of a singular integral operator

Summary

Consider a singular integral operator $a \varphi + b S \varphi + \int_\Gamma k \varphi$ on a closed curve $\Gamma$. Its index is the winding number around $0$ of the function $(a - b)/(a + b)$ on $\Gamma$. It is an integer computed from the coefficients alone, and it does not depend on the regular part $k$. By the Noether theorems, it equals the number of independent solutions of the homogeneous equation minus that of the adjoint homogeneous equation. So it is the Fredholm index of the operator. It decides how far the classical Fredholm alternative fails for the equation.