Transcendental Values of Some Elliptic Integrals

M. Ram Murty and Purusottam Rath
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
Purusottam Rath: Chennai Mathematical Institute

Chapter Chapter 13 in Transcendental Numbers, 2014, pp 59-63 from Springer

Abstract: Abstract In the case of trigonometric functions, we can rewrite the familiar identity sin 2 z + cos 2 z = 1 $$\displaystyle{\sin ^{2}z +\cos ^{2}z = 1}$$ as y 2 + d y d z 2 = 1 $$\displaystyle{y^{2} + \left ({dy \over dz}\right )^{2} = 1}$$ where y(z) = sinz. We can retrieve the inverse function of sine by formally integrating d z = d y 1 − y 2 , $$\displaystyle{dz ={ dy \over \sqrt{1 - y^{2}}},}$$ so that sin − 1 z = ∫ 0 z d y 1 − y 2 . $$\displaystyle{\sin ^{-1}z =\int _{ 0}^{z}{ dy \over \sqrt{1 - y^{2}}}.}$$ The period of the sine function can also be retrieved from 2 π = 4 ∫ 0 1 d y 1 − y 2 . $$\displaystyle{2\pi = 4\int _{0}^{1}{ dy \over \sqrt{1 - y^{2}}}.}$$ However, we should be cautious about this reasoning since sin−1 z is a multi-valued function and the integral may depend on the path taken from 0 to z. With this understanding, let us try to treat the inverse of the elliptic function ℘ ( z ) $$\wp (z)$$ in a similar way. Indeed, we have d ℘ ( z ) d z = 4 ℘ ( z ) 3 − g 2 ℘ ( z ) − g 3 $$\displaystyle{{d\wp (z) \over dz} = \sqrt{4\wp (z)^{3 } - g_{2 } \wp (z) - g_{3}}}$$

Keywords: Elliptic Curve; Elliptic Function; Trigonometric Function; Sine Function; Elliptic Integral (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-0832-5_13

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DOI: 10.1007/978-1-4939-0832-5_13

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