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Elliptic functions: Introduction course

Elliptic functions: Introduction courseVladimir G. TKACHEVD epartment of Mathematics, Royal Institute of TechnologyLindstedtsv agen 25, 10044 Stockholm, Swedenemail: tkatchevContentsChapter 1. Elliptic integrals and Jacobi s theta Elliptic integrals and the AGM: real Lemniscates and elastic Euler s addition Theta functions: preliminaries24 Chapter 2. General theory of doubly periodic Periods of analytic Existence of doubly periodic Liouville s The Weierstrass function (z) Modular forms51 Bibliography613 CHAPTER 1 Elliptic integrals and Jacobi s theta Elliptic integrals and the AGM: real Arclength of an ellipse with major and minor arcs 2aand2band eccentricitye:= (a2 b2)/a2 [0,1), ,x2a2+y2b2= is the arclength`(a;b) of the ellipse, as a function ofaandb?]

Elliptic integrals and Jacobi’s theta functions 1.1. Elliptic integrals and the AGM: real case 1.1.1. Arclength of ellipses. Consider an ellipse with major and minor arcs 2a and 2b and eccentricity e := (a2 −b2)/a2 ∈ [0,1), e.g., x2 a2 + y2 b2 = 1. What is the arclength `(a;b) of the ellipse, as a function of a and b? There are two easy

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