A Generalisation of the Cauchy Integral Formula for Normal Matrices

作者：Jefferies Brian^*

来源：Complex Analysis and Operator Theory, 2012, 6(5): 1037-1046.

DOI：10.1007/s11785-012-0226-x

• Cauchy integral formula
• Normal matrix
• Weyl functional calculus

The Cauchy integral formula says that %26lt;br%26gt;1/2 pi i integral f(z)/z-m dz = f(m) %26lt;br%26gt;if f is holomorphic in a neighbourhood U of m epsilon C and C is a simple Jordan curve contained in U about m. In this note, we express %26lt;br%26gt;1/2 pi i integral f(z)/det(zI-M) dz %26lt;br%26gt;as an average over the numerical range co(sigma(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 sigma(M) of eigenvalues of M. The expression is of use in determining the propagation cone of a symmetric hyperbolic system of PDE.

• 出版日期2012-10