Leonard Lewin (1919–2007): The Architect of Polylogarithms and Microwave Theory
Leonard Lewin was a rare breed of scholar who occupied the fertile middle ground between theoretical mathematics and practical electrical engineering. While his professional career was rooted in the development of microwave technology and telecommunications, his enduring intellectual legacy lies in his obsession with a specific family of mathematical functions: the polylogarithms. His work bridged the gap between the wartime necessity of radar and the abstract elegance of number theory.
1. Biography: From Radar to Boulder
Born on July 22, 1919, in Southend-on-Sea, England, Leonard Lewin’s early life was shaped by the rapid technological shifts of the interwar period. He received his education in London, earning a degree in mathematics from the University of London.
His career began in earnest during World War II, a period that forced many mathematicians into the applied sciences. Lewin joined the British company Standard Telephones and Cables (STC), a subsidiary of ITT, where he worked on the cutting edge of microwave research and radar development. This practical immersion in electromagnetic theory would provide the catalyst for his mathematical inquiries.
In 1966, Lewin transitioned from industry to academia, moving to the United States to join the faculty at the University of Colorado Boulder. He served as a Professor of Electrical Engineering and played a pivotal role in establishing the university’s reputation as a center for electromagnetics. He remained at Boulder for the rest of his career, eventually becoming a Professor Emeritus until his death on January 2, 2007.
2. Major Contributions: Engineering and "Recreational" Math
Lewin’s contributions can be divided into two distinct but overlapping spheres:
Microwave Engineering and Waveguides
Lewin was a pioneer in the study of waveguides—structures that guide electromagnetic waves with minimal loss of energy. He specialized in "discontinuities" within waveguides. When a waveguide changes shape or hits an obstacle, it creates complex mathematical problems. Lewin developed analytical methods to solve these electromagnetic equations, which were essential for the development of satellite communications and modern radar.
The Revival of Polylogarithms
Lewin’s most profound impact on mathematics was his revival of the polylogarithm ($\text{Li}_n(z)$), a function that had been largely ignored since the 18th and 19th centuries.
- The Dilogarithm ($\text{Li}_2$): Lewin focused heavily on the dilogarithm, discovering a vast array of "special values" and functional equations (identities that relate the function at different arguments).
- Bridge to Physics: While he initially treated these as a form of intellectual recreation, these functions later became indispensable in Quantum Electrodynamics (QED). When physicists calculate the interactions of subatomic particles using Feynman diagrams, the results almost invariably involve the polylogarithms Lewin mapped out decades earlier.
3. Notable Publications
Lewin was a prolific writer, authoring over 200 technical papers and several foundational books.
- Advanced Theory of Waveguides (1951): This established him as a leading voice in electromagnetic theory.
- Dilogarithms and Associated Functions (1958): This is considered his magnum opus. At the time of its publication, the dilogarithm was an obscure curiosity. Lewin’s systematic treatment turned it into a standard tool for mathematical analysis.
- Theory of Waveguides (1975): An updated, comprehensive text that became a standard reference for graduate students in electrical engineering.
- Polylogarithms and Associated Functions (1981): An expansion of his 1958 work, this book remains the "bible" for researchers working with these functions in number theory and particle physics.
- Structural Aspects of Polylogarithms (1991): An edited volume that brought together the world's leading mathematicians to discuss the deeper algebraic structures Lewin had spent a lifetime uncovering.
4. Awards & Recognition
Though Lewin did not seek the limelight, his peers recognized him as a titan in his field:
- IEEE Fellow: Elected for his contributions to the theory of waveguides and antenna design.
- W.G. Baker Award (1952): Awarded for his outstanding research in radio engineering.
- MTT-S Microwave Career Award (1993): The highest honor bestowed by the IEEE Microwave Theory and Techniques Society, recognizing a lifetime of "pioneer contributions" to the field.
- Honorary Doctorate: Awarded by the University of Colorado for his distinguished service and academic excellence.
5. Impact & Legacy
Lewin’s legacy is twofold. In the world of telecommunications, his work on waveguides helped lay the groundwork for the infrastructure of the digital age, from satellite uplinks to high-frequency circuit design.
In the world of mathematics and physics, Lewin is remembered as the man who kept the flame of polylogarithms alive. Today, these functions are central to:
- Number Theory: Specifically in the study of zeta functions and L-functions.
- High-Energy Physics: They are essential for calculating "loop integrals" in particle accelerators like the Large Hadron Collider.
- K-Theory: A branch of abstract algebra where polylogarithms play a structural role.
Without Lewin’s meticulous cataloging of identities in the 1950s, the progress of theoretical physics in the 1980s and 90s would likely have been significantly delayed.
6. Collaborations and Mentorship
At the University of Colorado, Lewin was known as a dedicated mentor. He collaborated extensively with colleagues such as David C. Chang and Edward F. Kuester on electromagnetic problems.
In his mathematical pursuits, he maintained a global correspondence with number theorists. His 1991 book, Structural Aspects of Polylogarithms, featured contributions from world-class mathematicians like Spencer Bloch and Don Zagier, illustrating how an engineer from Southend-on-Sea had earned the respect of the world’s most elite pure mathematicians.
7. Lesser-Known Facts
- The "Hobbyist" Mathematician: Lewin often referred to his work on polylogarithms as a "hobby." He would spend his evenings deriving complex identities that had no known application at the time, driven purely by an aesthetic appreciation for mathematical symmetry.
- Science and Religion: In his later years, Lewin became deeply interested in the philosophical intersection of science and faith. He wrote and lectured on how the complexity of the physical world could be reconciled with spiritual belief, reflecting a holistic view of human knowledge.
- The "Lewin Identities": In the mathematical literature, several highly complex identities involving the trilogarithm ($\text{Li}_3$) and higher orders are named after him. Some of these identities are so intricate that they were only verified years later through the use of advanced computer algebra systems.