The Swedish mathematician Gösta Mittag-Leffler (1846--1927) is well-known for founding Acta Mathematica, often touted as the first international journal of mathematics. A post-doctoral student in Paris and Berlin between 1873 and 1876, Mittag-Leffler built on Karl Weierstrass' work by proving the Mittag-Leffler Theorem, which states that a function of rational character (i.e. a meromorphic function) is specified by its poles, their multiplicities, and the coefficients in the principal part of its Laurent expansion. In this paper I explore the evolution of the Mittag-Leffler Theorem, from its initial state in 1876 to its final version, published in 1884. Mittag-Leffler's work contributed significantly to Weierstrass' program on the foundations of analysis. His interest in generalizing his results to functions having arbitrarily many essential singularities, however, which led to his research on infinite sets of singular points, attracted him to Georg Cantor's set-theoretic work. As we shall see, his use of Cantor's theorems and definitions was closely linked to his research efforts in the context of Weierstrass' program.
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