A Generalisation of the Cauchy Integral Formula for Normal Matrices

Abstract

The Cauchy integral formula says that $$\frac{1}{2{\pi} i} \int\limits_{C} \frac{f( z)}{z - m}\, d z = f(m)$$ if f is holomorphic in a neighbourhood U of \({m \in \mathbb{C}}\) and C is a simple Jordan curve contained in U about m. In this note, we express $$\frac{1}{2{\pi} i} \int\limits_{C} \frac{f( z)}{\det( z I - M)}\, d z$$ as an average over the numerical range co(σ(M)) of a normal matrix M, when f is holomorphic in a neighbourhood U of the numerical range of M and C is a simple Jordan curve contained in U about the set σ(M) of eigenvalues of M. The expression is of use in determining the propagation cone of a symmetric hyperbolic system of PDE.