Lemniscate elliptic functions

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle x 2 + y 2 = x , {\displaystyle x^{2}+y^{2}=x,} the lemniscate sine relates the arc length to the chord length of a lemniscate ( x 2 + y 2 )

2 = x 2 − y 2 . {\displaystyle {\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}.}

The lemniscate functions have periods related to a number ϖ = {\displaystyle \varpi =} 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) π = {\displaystyle \pi =} 3.141592..., ratio of perimeter to diameter of a circle. As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle \{(1+i)\varpi ,(1-i)\varpi \},} and are a special case of two Jacobi elliptic functions on that lattice, sl ⁡ z = sn ⁡ ( z ; − 1 ) , {\displaystyle \operatorname {sl} z=\operatorname {sn} (z;-1),} cl ⁡ z = cd ⁡ ( z ; − 1 ) {\displaystyle \operatorname {cl} z=\operatorname {cd} (z;-1)} . Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with fundamental periods { 2 ϖ , 2 ϖ i } . {\displaystyle {\bigl \{}{\sqrt {2}}\varpi ,{\sqrt {2}}\varpi i{\bigr \}}.}

The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function ℘ ( z ; a , 0 ) {\displaystyle \wp (z;a,0)} .

Lemniscate sine and cosine functions

Definitions The lemniscate functions sl and cl can be defined as the solution to the initial value problem:

d d z sl ⁡ z = ( 1 + sl 2 ⁡ z ) cl ⁡ z , d d z cl ⁡ z = − ( 1 + cl 2 ⁡ z ) sl ⁡ z , sl ⁡ 0 = 0 , cl ⁡ 0 = 1 , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {sl} z={\bigl (}1+\operatorname {sl} ^{2}z{\bigr )}\operatorname {cl} z,\ {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cl} z=-{\bigl (}1+\operatorname {cl} ^{2}z{\bigr )}\operatorname {sl} z,\ \operatorname {sl} 0=0,\ \operatorname {cl} 0=1,}

or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners { 1 2 ϖ , 1 2 ϖ i , − 1 2 ϖ , − 1 2 ϖ i } : {\displaystyle {\big \{}{\tfrac {1}{2}}\varpi ,{\tfrac {1}{2}}\varpi i,-{\tfrac {1}{2}}\varpi ,-{\tfrac {1}{2}}\varpi i{\big \}}\colon }

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