Lemniscate sine

Sinus lemniscatus sl (black) and Cosinus lemniscatus cl (blue), for comparison the sine normalized to sl (light gray)

The length s of the lemniscate arc from the origin correlates with the distance r from the point of the curve to the origin.
Each quadrant contains a quarter arc (the length ) of the lemniscate. The focal points are attached here .

{\ displaystyle {\ tfrac {\ varpi} {2}}}

{\ displaystyle (\ pm {\ tfrac {1} {\ sqrt {2}}} \, | \, 0)}

The lemniscate sine or sine lemniscatus (short sinlemn or ) is a special, by the mathematician Carl Friedrich Gauss introduced mathematical function . The lemniscatic sine corresponds to the function for the lemniscate that is the sine for the circle. The lemniscate cosine (short coslemn or ) is derived directly from starting. Both are historically the first so-called elliptical functions today . ${\ displaystyle \ mathrm {sl}}$ ${\ displaystyle \ mathrm {cl}}$ ${\ displaystyle \ mathrm {sl}}$

In 1796, the 19-year-old Gauss (in notes published only after his death) dealt with the question of how one can calculate the distance of the corresponding point on the curve from the coordinate origin from a given arc length of a lemniscate . Mathematically, this leads to the inverse function of the elliptic integral ${\ displaystyle s}$ ${\ displaystyle r \ in (-1,1)}$ ${\ displaystyle r = 0}$ ${\ displaystyle r = r (s)}$

{\ displaystyle s (r) = \ int _ {0} ^ {r} {\ frac {\ mathrm {d} \, \ rho} {\ sqrt {1- \ rho ^ {4}}}}.}

Gauss called this inverse function Sinus lemniscatus and called it , thus ${\ displaystyle \ mathrm {sl}}$

{\ displaystyle r = \ mathrm {sl} \, s}

Accordingly, he defined the cosine lemniscatus , where the length of the semicircle is the lemniscate, so ${\ displaystyle \ mathrm {cl} \, s = \ mathrm {sl} \, ({\ tfrac {\ varpi} {2}} - s)}$ ${\ displaystyle \ varpi}$

{\ displaystyle \ varpi = 2 \ int _ {0} ^ {1} {\ frac {\ mathrm {d} \, \ rho} {\ sqrt {1- \ rho ^ {4}}}} \ approx 2 { ,} 62205 \ 75542 \ 92119 \ 81046 \, 48395 \ 89891 \ ldots}

(Follow A062539 in OEIS )

With these terms, Gauss was guided by the analogy to the circular functions , because the sine is the inverse function of the integral

{\ displaystyle s (r) = \ int _ {0} ^ {r} {\ frac {\ mathrm {d} \, \ rho} {\ sqrt {1- \ rho ^ {2}}}}, \ qquad {\ mbox {and}} \ qquad \ varpi = 2 \ int _ {0} ^ {1} {\ frac {\ mathrm {d} \, \ rho} {\ sqrt {1- \ rho ^ {2}} }} = \ pi.}

so and . His further decisive idea was to define the functions and not only for real numbers, but to continue them into the complex . He then proved the periodicity relations ${\ displaystyle r = \ sin s}$ ${\ displaystyle \ cos s = \ sin ({\ tfrac {\ pi} {2}} - s)}$ ${\ displaystyle \ mathrm {sl}}$ ${\ displaystyle \ mathrm {cl}}$

{\ displaystyle \ mathrm {sl} \, (s + 2 \ varpi) = \ mathrm {sl} \, s, \ qquad \ mathrm {sl} \, (s + 2 \ mathrm {i} \ varpi) = \ mathrm {sl} \, s.}

In contrast to the sine, the lemniscatic sine has two periods and , likewise, the function . The lemniscate functions are therefore elliptical . Carl Gustav Jacobi introduced the Jacobian elliptical functions around 1830 and thus generalized the two lemniscate functions. ${\ displaystyle \ mathrm {sl}}$ ${\ displaystyle 2 \ varpi}$ ${\ displaystyle 2 \ mathrm {i} \ varpi}$ ${\ displaystyle \ mathrm {cl}}$