Titlebook: Elliptic Integrals and Elliptic Functions; Takashi Takebe Textbook 2023 The Editor(s) (if applicable) and The Author(s), under exclusive l

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,über die Weierstra?sche ? — Funktion,How many kinds of domains are there in a plane? Here by the word ‘domain’ we mean a connected open set (or, equivalently, an arcwise-connected open set).

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,Analysis: Ma? und Integration,In the previous chapter we proved that rational functions, rational functions of an exponential function and elliptic functions have addition theorems (algebraic addition formulae). Are there other functions which have algebraic addition formulae? The next Weierstrass–Phragmén theorem1 answers this question.

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Introduction,In this chapter, apart from establishing rigorous definitions and logic, we will survey the various themes in the main part of the book to get an overview of the theory.We also pick up several topics which we shall not deal with later, in order to show the breadth and depth of the theory of elliptic functions.

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The Arc Length of CurvesAs a matter of fact, this is nothing more than a paraphrase of the definition of 𝜋: ‘The number 𝜋 is the ratio of a circle’s circumference to its diameter’. However, if you pursue logical rigour, there are many gaps to be filled.

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Classification of Elliptic IntegralsIt is natural to call the former an . integral, but why call the latter ‘elliptic’, even though the curve is not an ellipse? In fact, today the word ‘elliptic integral’ is a general term used in the following sense.

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Jacobi’s Elliptic Functions on In this chapter we introduce elliptic functions as inverse functions of elliptic integrals..We use several convergence theorems in real analysis, which we cite in Appendix A.