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Lemniscate Function

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The lemniscate functions arise in rectifying the arc length of the lemniscate. The lemniscate functions were first studied by Jakob Bernoulli and Giulio Fagnano. A historical account is given by Ayoub (1984), and an extensive discussion by Siegel (1969). The lemniscate functions were the first functions defined by inversion of an integral

L=2aint_0^1(1-t^4)^(-1/2)dt,
(1)

which was first done by Gauss, who noticed that

1/(agm(1,sqrt(2)))=2/piint_0^1(dt)/(sqrt(1-t^4)),
(2)

where agm(a,b) is the arithmetic-geometric mean (Borwein and Bailey 2003, p. 13).

Define the inverse lemniscate functions as

phi(x)=sinlemn^(-1)x
(3)
=int_0^x(1-t^4)^(-1/2)dt
(4)
=x_2F_1(1/4,1/2;5/4;x^4)
(5)
phi^'(x)=coslemn^(-1)x
(6)
=int_x^1(1-t^4)^(-1/2)dt
(7)
=K(i)-F(sin^(-1)x,i)
(8)
=(Gamma^2(1/4))/(4sqrt(2pi))-x_2F_1(1/4,1/2;5/4;x^4),
(9)

where _2F_1(a,b;c;z) is a hypergeometric function, F(z,k) is an incomplete elliptic integral of the first kind, K(k) is an elliptic integral of the second kind, and

pi=L/a,
(10)

so that

x=sinlemnphi
(11)
x=coslemnphi^'.
(12)

Now, there is an identity connecting phi and phi^' since

phi(x)+phi^'(x)=L/(2a)=1/2pi,
(13)

so

sinlemnphi=coslemn(1/2pi-phi).
(14)

These functions can be written in terms of Jacobi elliptic functions,

u=int_0^(sd(u,k))[(1-k^('2)y^2)(1+k^2y^2)]^(-1/2)dy.
(15)

Now, if k=k^'=1/sqrt(2), then

u=int_0^(sd(u,1/sqrt(2)))[(1-1/2y^2)(1+1/2y^2)]^(-1/2)dy
(16)
=int_0^(sd(u,1/sqrt(2)))(1-1/4y^4)^(-1/2)dy.
(17)

Let t=y/sqrt(2) so dy=sqrt(2)dt,

u=sqrt(2)int_0^(sd(u,1/sqrt(2))/sqrt(2))(1-t^4)^(-1/2)dt
(18)
u/(sqrt(2))=int_0^(sd(u,1/sqrt(2))/sqrt(2))(1-t^4)^(-1/2)dt
(19)
u=int_0^(sd(usqrt(2),1/sqrt(2))/sqrt(2))(1-t^4)^(-1/2)dt,
(20)

and

sinlemnphi=1/(sqrt(2))sd(phisqrt(2),1/(sqrt(2))).
(21)

Similarly,

u=int_(cn(u,k))^1(1-t^2)^(-1/2)(k^('2)+k^2t^2)^(-1/2)dt
(22)
=int_(cn(u,1/sqrt(2)))^1(1-t^2)^(-1/2)(1/2+1/2t^2)^(-1/2)dt
(23)
=sqrt(2)int_(cn(u,1/sqrt(2)))^1(1-t^4)^(-1/2)dt
(24)
u/(sqrt(2))=int_(cn(u,1/sqrt(2)))^1(1-t^4)^(-1/2)dt
(25)
u=int_(cn(usqrt(2),1/sqrt(2)))^1(1-t^4)^(-1/2)dt,
(26)

and

coslemnphi=cn(phisqrt(2),1/(sqrt(2))).
(27)

We know

coslemn(1/2pi)=cn(1/2pisqrt(2),1/(sqrt(2)))=0.
(28)

But it is true that

cn(K,k)=0,
(29)

so

K(1/(sqrt(2)))=1/2sqrt(2)pi=1/(sqrt(2))pi
(30)
(Gamma^2(1/4))/(4sqrt(pi))=1/(sqrt(2))pi
(31)
L=api=(Gamma^2(1/4))/(2^(3/2)sqrt(pi))a.
(32)

By expanding (1-t^4)^(-1/2) in a binomial series and integrating term by term, the arcsinlemn function can be written

phi(x)=int_0^x(dt)/(sqrt(1-t^4))
(33)
=sum_(n=0)^(infty)((1/2)_nx^(4n+1))/(n!(4n+1))
(34)
=x_2F_1(1/4,1/2;5/4;x^4),
(35)

where (a)_n is a Pochhammer symbol (Berndt 1994).

Ramanujan gave the following inversion formula for phi(x). If

(thetamu)/(sqrt(2))=sum_(n=0)^infty((1/2)_nx^(4n+1))/(n!(4n+1)),
(36)

where

mu=(Gamma^2(1/4))/(2pi^(3/2))
(37)

is the constant obtained by letting x=1 and theta=pi/2, and

v=2^(-1/2)sd(mutheta),
(38)

then

(mu^2)/(2x^2)=csc^2theta-1/pi-8sum_(n=1)^infty(ncos(2ntheta))/(e^(2pin)-1)
(39)

(Berndt 1994).

Ramanujan also showed that if 0<theta<pi/2, then

-mu/(sqrt(2))sum_(n=0)^infty((1/2)_nv^(4n-1))/(n!(4n-1))=cottheta+theta/pi+4sum_(n=1)^infty(sin(2ntheta))/(2^(2pin)-1),
(40)
lnv+1/6pi-1/2ln2+sum_(n=1)^infty((1/4)_nv^(4n))/((3/4)_n4n)=ln(sintheta)+(theta^2)/(2pi)-2sum_(n=1)^infty(cos(2ntheta))/(n(e^(2pin)-1)),
(41)
1/2tan^(-1)v=sum_(n=0)^infty(sin[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi])
(42)
1/4cos^(-1)(v^2)=sum_(n=0)^infty((-1)^ncos[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi]),
(43)

and

(sqrt(2))/(4mu)sum_(n=0)^infty(2^(2n)(n!)^2)/((2n+1)!(4n+3))v^(4n+3)=(pitheta)/8-sum_(n=0)^infty((-1)^nsin[(2n+1)theta])/((2n+1)^2cosh[1/2(2n+1)pi])
(44)

(Berndt 1994).

A generalized version of the lemniscate function can be defined by letting 0<=theta<=pi/2 and 0<=v<=1. Write

2/3thetamu=int_0^v(dt)/(sqrt(1-t^6)),
(45)

where mu is the constant obtained by setting theta=pi/2 and v=1. Then

mu=(sqrt(pi))/(Gamma(2/3)Gamma(5/6)),
(46)

and Ramanujan showed

(4mu^2)/(9v^2)=csc^2theta-2/(pisqrt(3))+8sum_(n=1)^infty((-1)^(n-1)ncos(2ntheta))/(e^(pinsqrt(3))-(-1)^n)
(47)

(Berndt 1994).

See also

Arithmetic-Geometric Mean, Elliptic Function, Elliptic Integral, Gauss's Constant, Hyperbolic Lemniscate Function

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References

Ayoub, R. "The Lemniscate and Fagnano's Contributions to Elliptic Integrals." Arch. Hist. Exact Sci. 29, 131-149, 1984.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 245, and 247-255, 258-260, 1994.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Siegel, C. L. Topics in Complex Function Theory, Vol. 1. New York: Wiley, 1969.

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Lemniscate Function

Cite this as:

Weisstein, Eric W. "Lemniscate Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LemniscateFunction.html

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