In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler:

Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.

I am interested in learning more about the context of this quote; why would Weierstrass have forbidden this technique? The article also states:

There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

So despite his initial objections, I guess Mittag-Leffler came around to the idea eventually?


"The master forbids it" Why would Weierstrass be thought to resist Borel summation of divergent series.

Well, two great predecessors of that era Euler and Riemann had used methods that were questioned: Euler used arbitrary series with great effect. Riemann characterized and constructed analytic functions using their singularities and geometric methods with great effect. To better understand mathematicians wanted perfectly transparent and rigorous methods. Weierstrass apparently led the way. He introduced criteria to test convergence and his notion of analytic continuation of convergent power series gave a satisfying picture. In particular it was eventually understood to be independent of coordinates.

So the answer to the question is "if its not broken don't fix it." It here being a coherent theory of analytic functions.

On the other hand, people work on generalizations of analytic functions which are analtyic in the Weierstrass sense on open sets and still have meaning on almost all lines even those passing through the closed set of singularities. Borel series can give examples of this .

In another realm they and their cousins help in the study of the formal series that arise in physics models related to quantum theory.

Dennis Sullivan


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