Lemniscatic elliptic function

In mathematics, a lemniscatic elliptic function is an elliptic function related to the arc length of a lemniscate of Bernoulli studied by Giulio Carlo de' Toschi di Fagnano in 1718. It has a square period lattice and is closely related to the Weierstrass elliptic function when the Weierstrass invariants satisfy $g 2 = 1$ and $g 3 = 0$ .

In the lemniscatic case, the minimal half period $ω 1$ is real and equal to

{\frac {\Gamma ^{2}\left({\frac {1}{4}}\right)}{4{\sqrt {\pi }}}}

where $Γ$ is the gamma function. The second smallest half period is pure imaginary and equal to $iω 1$ . In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

The constants $e 1$ , $e 2$ , and $e 3$ are given by

e_{1}={\tfrac {1}{2}},\qquad e_{2}=0,\qquad e_{3}=-{\tfrac {1}{2}}.

The case $g 2 = a$ , $g 3 = 0$ may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: $a > 0$ and $a < 0$ . The period parallelogram is either a square or a rhombus.

Lemniscate sine and cosine functions

The lemniscate sine (Latin: sinus lemniscatus) and lemniscate cosine (Latin: cosinus lemniscatus) functions $sinlemn$ aka $sl$ and $coslemn$ aka $cl$ are analogues of the usual sine and cosine functions, with a circle replaced by a lemniscate. They are defined by

\operatorname {sl} (r)=s

where

r=\int _{0}^{s}{\frac {dt}{\sqrt {1-t^{4}}}}

and

\operatorname {cl} (r)=c

where

r=\int _{c}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}.

They are doubly periodic (or elliptic) functions in the complex plane, with periods $2 π G$ and $2 π iG$ , where Gauss's constant $G$ is given by

G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}=0.8346\ldots .

Arclength of lemniscate

A lemniscate of Bernoulli and its two foci

The lemniscate of Bernoulli

\left(x^{2}+y^{2}\right)^{2}=x^{2}-y^{2}

consists of the points such that the product of their distances from the two points $(1 / \sqrt 2, 0)$ , $(- 1 / \sqrt 2, 0)$ is the constant 1/2. The length $r$ of the arc from the origin to a point at distance $s$ from the origin is given by

r=\int _{0}^{s}{\frac {dt}{\sqrt {1-t^{4}}}}.

In other words, the sine lemniscatic function gives the distance from the origin as a function of the arc length from the origin. Similarly the cosine lemniscate function gives the distance from the origin as a function of the arc length from (1, 0).

References

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 658. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Reinhardt, W. P.; Walker, P. L. (2010), "Lemniscate lattice", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Siegel, C. L. (1969). "Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory". Interscience Tracts in Pure and Applied Mathematics. 25. New York-London-Sydney: Wiley-Interscience A Division of John Wiley & Sons. ISBN 0-471-60844-0. MR 0257326.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Lemniscate functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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Lemniscatic elliptic function

Lemniscate sine and cosine functions

Arclength of lemniscate

See also

References

External links