Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 1037–1046 | Cite as

A Generalisation of the Cauchy Integral Formula for Normal Matrices

  • Brian JefferiesEmail author


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.


Cauchy integral formula Normal matrix Weyl functional calculus 


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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.UNSWSydneyAustralia

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