Graziano Gentili - Regular Functions of a Quaternionic Variable

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The theory of slice regular functions (originally Cullen-regular functions) was born at George Mason University, in Virginia, where the first and third authors collaborated for a 2-month period in the Fall of 2005. It was originated by the desire to find a new class of quaternionic regular functions that included polynomials and power series. The second author started working on this subject in the Summer of 2006, and her doctoral thesis eventually became the skeleton for this monograph. The theory of slice regular functions has rapidly developed, thanks to a series of visits at Chapman University, in California and to the interest of many mathematicians to whom we are greatly indebted.

We are very grateful to Fabrizio Colombo and Irene Sabadini, who immediately realized that this theory could be applied to create a successful quaternionic functional calculus. They also suggested the extension of these ideas to the case of Clifford algebras and their impulse has greatly contributed to the development of the theory.

We would like to thank Riccardo Ghiloni and Alessandro Perotti, who took an active interest in these developments and introduced a new viewpoint on the theory itself.

We warmly thank Cinzia Bisi, Alberto Damiano, Chiara Della Rocchetta, Giulia Sarfatti, Irene Vignozzi, and Fabio Vlacci for their interest in the subject and for their researches, which contributed to the expansion of the theory presented in this book.

We should also express our gratitude to Michael (Misha) V. Shapiro and to Mara Elena LunaElizarrars for their discussions with us and especially to Misha for his help in crafting a new introduction to this work, which better represents its relationship with other lines of research in the quaternionic field.

Special thanks go to Simon Salamon for his role in an unexpected application to the construction and classification of orthogonal complex structures in the quaternionic space.

Last but not least, we want to express our gratitude to the institutions who have supported us with the time needed for this work and in many cases have granted travel or local living expenses to the three of us. We gratefully acknowledge the support of: Chapman University, where most of the work has been done; George Mason University; Universit degli Studi di Firenze; Universit degli Studi di Milano; GNSAGA of the Istituto Nazionale di Alta Matematica F. Severi; European Social Fund; Regione Lombardia; MIURItalian Ministry of University and Researchvia the projects PRIN Propriet geometriche delle variet reali e complesse, PRIN Geometria Differenziale e Analisi Globale, and FIRB Geometria Differenziale Complessa e Dinamica Olomorfa.