Throughout his career Ahlfors made decisive contributions to several areas of complex analysis. Among his earliest results was the proof of the
Denjoy–Carleman–Ahlfors theorem, which states that the number of asymptotic values approached by an entire function of order ρ along curves in the
complex plane going toward infinity is less than or equal to 2ρ. His book
Complex Analysis (1953) is the classic text on the subject and is almost certainly referenced in any more recent text which makes heavy use of complex analysis. Ahlfors wrote several other significant books, including
Riemann surfaces (1960) and
Conformal invariants (1973). He made decisive contributions to
meromorphic curves,
value distribution theory,
Riemann surfaces,
conformal geometry,
quasiconformal mappings and other areas during his career. In 1954 Ahlfors proved that the results and conjectures of
Oswald Teichmüller — whose pioneering work on
Riemann surfaces had been cut short when he disappeared on the German Eastern Front in 1943 — were correct. In doing so he defined the concept of
Teichmüller space, which rapidly became an important field of research within function theory and later acquired significance in physics as well. His work made the theory of quasiconformal mappings a central area of complex analysis, and by the year 2000 this theory was assessed as perhaps the most important advance in function theory during the 20th century. == Personal life ==