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On the Theory of Maass Wave Forms

  • Textbook
  • © 2020


  • Provides a rigorous analytical treatment of the theory of Maass wave forms
  • Presents a self-contained exploration of the topic, from a brief review of introductory concepts through to cutting edge research
  • Offers readers a number of insightful exercises with solutions available online
  • Includes supplementary material:

Part of the book series: Universitext (UTX)

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About this book

This textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field.

Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online.

On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.

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Table of contents (7 chapters)


“The book is essentially self-contained; the only prerequisites are basic knowledge of complex analysis, real analysis, and abstract algebra. For the convenience of the reader, the book includes an appendix with background material. … The main use of the book is probably for advanced master's or graduate students to learn about the theory of Maass wave forms. However, the book should also be useful for established researchers in mathematics or related disciplines to enter the field.” (Sander Zwegers, Mathematical Reviews, October, 2022)

“The level of the textbook is suitable for master students and so on, but some proofs can easily be understood by advanced undergraduate students.” (Ilker Inam, zbMATH 1456.11002, 2021)

Authors and Affiliations

  • Department of Mathematics and Computer Science, FernUniversität in Hagen, Hagen, Germany

    Tobias Mühlenbruch

  • Department of Mathematics, American University of Beirut, Beirut, Lebanon

    Wissam Raji

About the authors

Tobias Mühlenbruch is a private lecturer at the FernUniversität in Hagen, Germany. His main research interest include number theory with a focus on analytic modular forms and the connections to dynamical systems and mathematical physics. 

Wissam Raji is an Associate Professor of Mathematics at the American University of Beirut, Lebanon. His research interests include analytic number theory and modular forms with a focus on period polynomials. He is the recipient of numerous awards, including the Saylor Foundation’s 2013 Open Textbook Challenge.

Bibliographic Information

  • Book Title: On the Theory of Maass Wave Forms

  • Authors: Tobias Mühlenbruch, Wissam Raji

  • Series Title: Universitext

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2020

  • Softcover ISBN: 978-3-030-40477-2Published: 12 June 2020

  • eBook ISBN: 978-3-030-40475-8Published: 06 May 2020

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XXV, 509

  • Number of Illustrations: 11 b/w illustrations

  • Topics: Number Theory

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