This book presents a scholarly English translation of Henri Lebesgue’s seminal 1909 paper, Sur les Intégrales Singulières, first published in the Annales de l’Université de Toulouse. This foundational work, long known to specialists in French, is made accessible here to a broader mathematical audience through a clear, faithful and pedagogically enriched translation. In addition to the core translation, the book features an original introductory chapter and an appendix by the translator that trace the historical development of harmonic analysis from classical trigonometric series to the Zygmund–Calderón theory, with mention of Carleson’s breakthrough in almost everywhere convergence.

The historical survey outlines the evolution of ideas contributed by Dirichlet, Fejér, Dini, Vallée Poussin, Jordan, Kolmogorov, Young and others, illuminating the pathway from early convergence results to modern singular integral theory. The material is presented in structured chapters and enhanced with diagrams, worked examples and graded exercises, offering an accessible yet rigorous learning experience. Supplementary readings are included to encourage deeper exploration and contextual understanding.

This volume is ideal for advanced undergraduate and graduate students, as well as researchers interested in Fourier analysis, trigonometric series and the early development of harmonic analysis. It provides a rich conceptual bridge between classical analysis and modern harmonic tools. It is particularly well-suited for courses on harmonic analysis and the historical foundations of integration theory. Mathematicians, educators and historians of mathematics will find in this book both a pedagogical resource and a valuable preservation of Lebesgue’s contribution to the analytic tradition.

Rahul Jain, Ph.D., is Director at the National Security Council Secretariat in New Delhi, India. His academic journey commenced at Kurukshetra University, Haryana, India, where he completed his undergraduate studies. He later pursued postgraduate and doctoral studies at the Indian Institute of Science (IISc) and the Tata Institute of Fundamental Research Centre for Applied Mathematics (TIFR-CAM), Bengaluru, Karnataka, India, from 2001 to 2006. His research contributions have been recognised and published in esteemed journals, including in Computers and Mathematics with Applications. These publications underscore his proficiency in addressing complex mathematical problems with rigor and precision.

Actively participating in academic events, he attended Trimester Schools in Italy and Portugal, presenting a paper in France on “Propagation and cancellation of singularities in a class of Fuchsian operators and their perturbations.” Dedicated to knowledge dissemination, he has served as a teaching assistant at both IISc and TIFR-CAM, guiding students in their mathematical journeys. He delivered lectures at Rajasthan University, Jaipur, fostering a passion for mathematics among learners. Proficient in various branches of mathematics, he enjoys making intricate theories more accessible to students. Fluent in French, he is involved in translating classical French mathematics books for non-French-speaking students worldwide.