The Hankel transform, a mathematical operation that integral transforms a function, stands as a primary contribution from a short but highly productive career in 19th-century analysis. Born in 1839 in Halle, Germany, Hermann Hankel spent his academic life refining complex number theory, matrix theory, and functional analysis before his death in 1873 at age 34.
Academic Formation
Hankel completed his early studies at the Nicolai Gymnasium in Leipzig before entering Leipzig University in 1857. He subsequently expanded his mathematical expertise at the University of Göttingen in 1860, where he focused on function theory under Bernhard Riemann. By 1861, he moved to Frederick William University Berlin to study with Karl Weierstrass and Leopold Kronecker, ultimately receiving his doctorate from Leipzig University in 1862.
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After earning his teaching qualifications in 1863, Hankel advanced through several German institutions. He served at Leipzig University from 1863 to 1867, achieving the rank of associate professor. His career continued with a full professorship at the University of Erlangen–Nuremberg from 1867 to 1869. He concluded his final four years as a professor at the University of Tübingen, where he resided until his death in Schramberg.
Contributions to Mathematical Analysis
Hankel’s research fundamentally influenced complex analysis and the study of functions. His 1867 treatise, Theorie der Complexen Zahlensysteme, explored quaternions and complex numbers, providing a rigorous proof for the rule of signs in multiplication. His later work, including the 1870 Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen and his contribution to the Ersch-Gruber Encyklopädie, highlighted the importance of Bessel functions, which were subsequently designated as Hankel functions.
Historical and Theoretical Rigor
Beyond his work in analysis, Hankel engaged deeply with the history and structure of mathematics. He promoted the extension theory developed by Hermann Grassmann, effectively utilizing these insights to analyze the nature of space. His scholarly output also included Die Entwickelung der Mathematik in den letzten Jahrhunderten, published in 1869, which examined the evolution of mathematical thought.
Fast facts
- Born: 1839, Halle (Saale), Germany
- Died: 1873, Schramberg, Germany
- Citizenship: Kingdom of Prussia
- Education: Leipzig University, University of Göttingen, Frederick William University Berlin
- Primary Fields: Mathematical analysis, number theory, matrix theory
- Notable Works: Hankel transform, Hankel matrix, Hankel function, Hankel contour
- Academic Tenure: Leipzig, Erlangen–Nuremberg, Tübingen
- Award: Honorary member of the Physics Association, Frankfurt am Main (1857)
Questions readers ask
What is the primary legacy of Hermann Hankel?
He is recognized for introducing the Hankel transform, the Hankel matrix, and the Hankel function, which are essential tools in modern mathematical analysis.
Where did Hankel hold professorships during his lifetime?
He was an associate professor at Leipzig University and held full professorships at the University of Erlangen–Nuremberg and the University of Tübingen.
Achievements
- Notable work: Hankel matrix
- Notable work: Hankel function
- Notable work: Hankel transform
- Notable work: spherical Hankel function
- Held posts at University of Tübingen, Leipzig University and University of Erlangen–Nuremberg
- Fields: mathematical analysis, special function and mathematics
