Transcription of Elliptic functions: Introduction course
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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?]
Of course, π is transcendental so it is debatable how well we understand it! –1 –0.5 0 0.5 1 –2 –1 1 2 Figure 1. Ellipse x2 + y2 4 = 1 The total arclength is four times the length of the piece in the first quadrant, where we have the relations y = b p 1−(x/a)2, y0(x) = − xb a2 1 p 1−(x/a)2. 5
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