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Regularized trace of the inverse of the dirichlet laplacian
Author(s) -
Dostanić Milutin R.
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20368
Subject(s) - mathematics , trace (psycholinguistics) , bounded function , dirichlet distribution , inverse , laplace operator , eigenvalues and eigenvectors , dirichlet eigenvalue , convex domain , pure mathematics , mathematical analysis , dirichlet's principle , physics , geometry , boundary value problem , philosophy , linguistics , quantum mechanics
For the eigenvalues $( \lambda_{n}) _{n=1}^{\infty}$ of the Dirichlet Laplacian on a bounded convex domain $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\Omega\subset{\C}$ , we find the sum of the series$$\sum\limits_{n=1}^{\infty} \left( {{1}\over{\lambda_{n}}}-{{\vert \Omega\vert }\over{4\pi n}}\right),$$ the regularized trace of the inverse of Dirichlet Laplacian. © 2011 Wiley Periodicals, Inc.