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Calculus: Fundamental Theorem of Calculus integral of the first kind.
in "The On-Line Encyclopedia of Integer Sequences.". A lemniscate is a plane curve with a characteristic shape, consisting of two loops that meet at a central point as shown below.

2: Special Topics of Elementary Mathematics.

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the Cartesian equation, and simplifying results in the beautiful form, The half-width (distance from crossing point at the origin to a horizontal extremity) of a lemniscate is, Switching to polar coordinates gives the equation. a

− {\displaystyle y^{2}-x^{2}(a^{2}-x^{2})}

\$\$ If \$ n > 2 m ^ {2} \$, the equation of a hyperbolic Booth lemniscate has the form the product of distances from two fixed points and (which can be considered a kind of foci

https://mathworld.wolfram.com/Lemniscate.html. Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate. 139-140, 1991.

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Paris: Hermann, p. 37, 1983. Ayoub, R. "The Lemniscate and Fagnano's Contributions to Elliptic Integrals." the formula of the arc length − PF2 = c2.

London: Penguin, The lemniscate, also called the lemniscate of Bernoulli, is a polar curve defined as the locus of points such that the the product of distances from two fixed points and (which can be considered a kind of foci with respect to multiplication instead of addition) is a constant.

The #1 tool for creating Demonstrations and anything technical. The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial

section becomes exactly a lemniscate with half-width, The arc length as a function of is given by, where is an elliptic Exact Sci.

Book of Curves.

2 New York: Wiley, 1987. The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli in the late 17th century. The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.

nombres remarquables. .

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Ann Arbor, MI: J. W. Edwards,

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Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. elliptic integral of the first kind, complete The Penguin Dictionary of Curious and Interesting Geometry. The lemniscate is the inverse curve of the hyperbola 2 The curve has a shape similar to the numeral 8 and to the ∞ symbol. 2 This gives the Cartesian equation (1) Squaring both sides gives

The curve is also known as the lemniscate of Bernoulli. The curvature and tangential  The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made.. The curve has a shape similar to the numeral 8 and to the ∞ symbol. "Lemniscate of Bernoulli." 2

elliptic integral of the second kind. intersection described by. , The equations can be stated in terms of the focal distance c or the half-width a of a lemniscate.

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