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Hyperbolic Lemniscate Function


By analogy with the lemniscate functions, hyperbolic lemniscate functions can also be defined

arcsinhlemnx=int_0^x(1+t^4)^(1/2)dt
(1)
=x_2F_1(-1/2,1/4;5/4;-x^4)
(2)
arccoshlemnx=int_x^1(1+t^4)^(1/2)dt
(3)
=_2F_1(-1/2,1/4;5/4;-1)-x_2F_1(-1/2,1/4;5/4;-x^4).
(4)

where _2F_1(a,b;c;z) is a hypergeometric function.

Let 0<=theta<=pi/2 and 0<=v<=1, and write

(thetamu)/2=int_0^v(dt)/(sqrt(1+t^4))
(5)
=v_2F_1(1/4,1/2;5/4;-v^4),
(6)

where mu is the constant obtained by setting theta=pi/2 and v=1, which is given by

mu=2/piK(1/(sqrt(2)))
(7)
=(sqrt(pi))/(Gamma^2(3/4)),
(8)

with K(k) is a complete elliptic integral of the first kind. Ramanujan showed that

 2tan^(-1)v=theta+sum_(n=1)^infty(sin(2ntheta))/(ncosh(npi)),
(9)
 1/8pi-1/2tan^(-1)(v^2)=sum_(n=0)^infty((-1)^ncos[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi])
(10)

and

 ln((1+v)/(1-v))=ln[tan(1/4pi+1/2theta)]+4sum_(n=0)^infty((-1)^nsin[(2n+1)theta])/((2n+1)[e^((2n+1)pi)-1])
(11)

(Berndt 1994).


See also

Lemniscate Function

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References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 255-258, 1994.

Referenced on Wolfram|Alpha

Hyperbolic Lemniscate Function

Cite this as:

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

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