The Schwarz function and its generalization to higher dimensions.

*(English)*Zbl 0784.30036
The University of Arkansas Lecture Notes in the Mathematical Sciences. New York: John Wiley & Sons Ltd. xiv, 108 p. (1992).

The so-called Schwarz function comes from the following result. Proposition: Let \(\Gamma\) be a nonsingular analytic Jordan arc in \({\mathbf R}^ 2\cong{\mathbf C}\). Then there are a neighborhood \(\Omega\) of \(\Gamma\) and a unique holomorphic function \(S\) on \(\Omega\) such that \(S(z)=\bar z\) for \(z\in \Gamma\). Shades of Plemelj! Clearly, this result is closely related to (anti-holomorphic) reflection.

In this little volume, the author weaves a delightful tapestry using this result as his touchstone. He treats reflection principles, Plemelj formulas, analytic continuation, Dirichlet problems, Neumann problems, and Cauchy problems. He looks at both the one- and several-variable theories. He treats classical results on reproducing formulas in a new way.

In all, the reader is treated (in a scant 96 pages) to a panorama of important issues in modern analysis. Since reflection principles are currently enjoying a surge in popularity, the book should be of particular interest.

In this little volume, the author weaves a delightful tapestry using this result as his touchstone. He treats reflection principles, Plemelj formulas, analytic continuation, Dirichlet problems, Neumann problems, and Cauchy problems. He looks at both the one- and several-variable theories. He treats classical results on reproducing formulas in a new way.

In all, the reader is treated (in a scant 96 pages) to a panorama of important issues in modern analysis. Since reflection principles are currently enjoying a surge in popularity, the book should be of particular interest.

Reviewer: St.Krantz (St.Louis)

##### MSC:

30E99 | Miscellaneous topics of analysis in the complex plane |

30C40 | Kernel functions in one complex variable and applications |

30E25 | Boundary value problems in the complex plane |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |