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Lindelöf spent the year 1891 in Stockholm, the years 1893-94 in Paris and 1901 in Göttingen. He returned to Helsingfors where he became assistant professor, becoming a full professor the following year. He remained in that post until he retired in 1938. From 1907 he served on the editorial board of Acta Mathematica.
Lindelöf's first work in 1890 was on the existence of solutions for differential equations. It is an outstanding paper. Then he worked on analytic functions, applying results of Mittag-Leffler in a study of the asymptotic investigation of Taylor series. In particular he was interested in the behaviour of such functions in the neighbourhood of singular points.
He considered analogues of Fourier series and applied them to gamma functions. He also wrote on conformal mappings. His work on analytic continuation is explained in a well-written book Le calcul des résidus et ses applications à la théorie des fonctions (Paris, 1905).
He gave up research to devote himself to teaching and writing excellent textbooks.
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