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The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine y = sin(πx/ϖ) (pale dashed red).. In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli.
The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae).
One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice can be expressed as a rational function in terms of ℘ and ℘ ′. [7] The ℘-function satisfies the differential equation
The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate. Lemniscate of Gerono
Salk Institute researchers have identified a key protein that may help users of GLP-1 drugs such as Ozempic and Wegovy maintain muscle mass while losing weight.
The fruit also contains nutrients and other bioactive compounds that can support sexual function and may even help with issues such as erectile dysfunction, or ED, as part of an overall healthy ...
The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. [2] It also appears in evaluation of the gamma and beta function at certain rational values.