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SOME PROPERTIES OF THE CURVES xn+yn = 1 WITH EVEN EXPONENTS

SOME PROPERTIES OF THE CURVES xn+yn = 1 WITH EVEN EXPONENTS DEMONSTRATIO MATHEMATICAVol. XLIIINo 12010Irena FidytekSOME PROPERTIES OF THE CURVES xn + yn = 1WITH EVEN E X P O N E N T SAbstract. We present parametric equations for the curve xn + yn = 1 with evenexponents, in terms of the double area of the sector bounded by the arc between (1,0) and(x, y) and the radius vectors of these points. We determine also the area enclosed by thiscurve.1. IntroductionThe curves xn + yn = 1 with even exponents have been studied by different authors in different aspects. In [1] R. Tardiff has examined thesecurves with regards on some geometrical properties. A. Levin studying thecurve x4 + y4 = 1 in [2] found infinite product for the lemniscate constantL corresponding to the Viete's product for IT. Moreover he defined somefunctions c(ip), s(<p) parametrizing the curve x4 + y4 = 1 and such that thearea of the sector between the points (0,0), (1,0) and (c(tp), s((p)) is equalto ^¡p. These functions yield some analogues of the classical sine and cosinefunctions. The author showed also, that the squares of the function c(ip)and s((p) may be expressed with the aid of Weierstrass' gamma function;moreover, some addition formulas for these functions are http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SOME PROPERTIES OF THE CURVES xn+yn = 1 WITH EVEN EXPONENTS

Fidytek, Irena
Demonstratio Mathematica , Volume 43 (1): 8 – Jan 1, 2010
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References (3)

Publisher
de Gruyter
Copyright
© 2017 by Irena Fidytek
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2010-0109
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XLIIINo 12010Irena FidytekSOME PROPERTIES OF THE CURVES xn + yn = 1WITH EVEN E X P O N E N T SAbstract. We present parametric equations for the curve xn + yn = 1 with evenexponents, in terms of the double area of the sector bounded by the arc between (1,0) and(x, y) and the radius vectors of these points. We determine also the area enclosed by thiscurve.1. IntroductionThe curves xn + yn = 1 with even exponents have been studied by different authors in different aspects. In [1] R. Tardiff has examined thesecurves with regards on some geometrical properties. A. Levin studying thecurve x4 + y4 = 1 in [2] found infinite product for the lemniscate constantL corresponding to the Viete's product for IT. Moreover he defined somefunctions c(ip), s(<p) parametrizing the curve x4 + y4 = 1 and such that thearea of the sector between the points (0,0), (1,0) and (c(tp), s((p)) is equalto ^¡p. These functions yield some analogues of the classical sine and cosinefunctions. The author showed also, that the squares of the function c(ip)and s((p) may be expressed with the aid of Weierstrass' gamma function;moreover, some addition formulas for these functions are

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 2010

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