Mathematical Sciences: Square Summable Power Series

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deBranges 9622445 In 1984 the present investigator obtained a proof of a conjecture, made in 1916 by Ludwig Bieberbach, estimating the coefficients of power series which represent injective analytic functions in the unit disk. The proof was included in a manuscript on square summable power series which motivated the argument and prepared the way for applications. The verification on the argument required an extensive visit to the then Soviet Union during which the proof was separated from its original context. A dim perception of the proof was conveyed which was insufficient as a guide for further research. The need for such research was even obscured by the commonly accepted view that an era had come to a close. The research opportunities generated by the proof were underestimated. The need of an improved treatment of the theory of square summable power series became evident. The research required for such a treatment is the aim of the present project. The significance of mathematics for the global economy is difficult to assess because it proceeds through intermediaries in science and engineering. Square summable power series are an aspect of complex analysis, and also of functional analysis, which is most closely related to engineering through the theory of linear systems. The interplay is seen in proofs of the Bieberbach conjecture which appear in engineering journals. An interplay at a higher level is seen in proofs of the Bieberbach conjecture which appear in advanced mathematical texts. The technical and scientific community has in this way expressed its need for information about the proof. But it has at the same time exposed the inadequacy of existing knowledge of the proof even among experts. A substantial effort is required from the author of the proof to surmount these difficulties.